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A numerical approach to optimization problems with variational inequality constraints
Abstract
Optimization problems with variational inequality constraints are converted to constrained minimization of a local Lipschitz function. To this minimization a non-differentiable optimization method is used; the required subgradients of the objective are computed by means of a special adjoint equation. Besides tests with some academic examples, the approach is applied to the computation of the Stackelberg—Cournot—Nash equilibria and to the numerical solution of a class of quasi-variational inequalities.
... As before, it is assumed that the non-smoothness, which here occurs in the semi-linear Non-smooth optimization problems with an elliptic PDE constraint, which involves the mentioned non-smooth non-linear functions, arise in many modern applications. They may arise in mathematical models representing physical systems and engineering problems, e.g., areas of signal and image processing, mechanics and plasma physics, as well as robotics; see e.g., [16,23,70,71,118,161,166,175,192]. For brevity, only two representative applications are mentioned here. ...
... Hence, many problems of great practical relevance fall into this category, e.g., the control of Bingham fluids or of (thermo-)plastic deformations, contact or free boundary problems, as well as economics, 5.7. On Solving Obstacle Problems with CALi see, e.g., [16,33,42,73,105,106,122,119,161,187,189]. Due to the VI constraint these kinds of problems are non-smooth and non-convex which complicates their theoretical and numerical treatment. ...
Nichtglatte Optimierungsprobleme in reflexiven Banachräumen treten in vielen Anwendungen auf. Häufig wird angenommen, dass alle vorkommenden Nichtdifferenzierbarkeiten durch Lipschitz-stetige Operatoren wie abs, min und max gegeben sind. Bei solchen Problemen kann es sich zum Beispiel um optimale Steuerungsprobleme mit möglicherweise nicht glatten Zielfunktionen handeln, welche durch partielle Differentialgleichungen (PDG) eingeschränkt sind, die ebenfalls nicht glatte Terme enthalten können. Eine effiziente und robuste Lösung erfordert eine Kombination numerischer Simulationen und spezifischer Optimierungsalgorithmen. Lokal Lipschitz-stetige, nichtglatte Nemytzkii-Operatoren, welche direkt in der Problemformulierung auftreten, spielen eine wesentliche Rolle in der Untersuchung der zugrundeliegenden Optimierungsprobleme. In dieser Dissertation werden zwei spezifische Methoden und Algorithmen zur Lösung solcher nichtglatter Optimierungsprobleme in reflexiven Banachräumen vorgestellt und diskutiert. Als erste Lösungsmethode wird in dieser Dissertation die Minimierung von nichtglatten Operatoren in reflexiven Banachräumen mittels sukzessiver quadratischer Überschätzung vorgestellt, SALMIN. Ein neuartiger Optimierungsansatz für Optimierungsprobleme mit nichtglatten elliptischen PDG-Beschränkungen, welcher auf expliziter Strukturausnutzung beruht, stellt die zweite Lösungsmethode dar, SCALi. Das zentrale Merkmal dieser Methoden ist ein geeigneter Umgang mit Nichtglattheiten. Besonderes Augenmerk liegt dabei auf der zugrundeliegenden nichtglatten Struktur des Problems und der effektiven Ausnutzung dieser, um das Optimierungsproblem auf angemessene und effiziente Weise zu lösen.
... The symbol '/' means that the number of iterations exceeds 10,000 or the CPU time exceeds 10 s. Example 1: This example is tested by Harker [17], Outrata and Zowe [44], Zhang et al. [37] and Han et al. [38]. It is a two-person game, in which each player picks a number x i between 0 and 10 and the sum of their numbers must be less than or equal to 15. ...
... Thus x * ∈ S * . Example 3: This example is a modification of the Stackelberg-Cournot-Nash equilibrium problem in [44] and tested by Zhang et al. [37] and Han et al. [38]. Consider an oligopolistic market in which m firms supply a homogeneous product in a noncooperative fashion. ...
The generalized Nash equilibrium problem (GNEP) is an n-person noncooperative game in which each player’s strategy set depends on the rivals’ strategy set. In this paper, we presented a half-space projection method for solving the quasi-variational inequality problem which is a formulation of the GNEP. The difference from the known projection methods is due to the next iterate point in this method is obtained by directly projecting a point onto a half-space. Thus, our next iterate point can be represented explicitly. The global convergence is proved under the minimal assumptions. Compared with the known methods, this method can reduce one projection of a vector onto the strategy set per iteration. Numerical results show that this method not only outperforms the known method but is also less dependent on the initial value than the known method.
... , F m ) T : R n+m → R m are continuously differentiable, and a⊥b denotes orthogonality of any vectors a, b ∈ R n , i.e., a ⊤ b = 0. Such problems play an important role in many fields such as the design of transportation networks, economic equilibrium and engineering design, see [1,2]. ...
... When is a subset of the natural space for the state variable, this class of problems is often referred to as Minimization Problems with Equilibrium Constraints (MPEC) (Luo (1a) min et al. 1996;Kočvara Outrata 1995;Outrata 1994;Outrata and Zowe 1995;Baumrucker et al. 2008;Maury et al. 2018). In this case, the first-order optimality conditions for (1b) take the form of the system of Karush-Kuhn-Tucker (KKT) conditions which can be written as variational inequalities of the first kind (Kinderlehrer and Stampacchia 2000), and the classical adjoint approach for PDE-constrained optimization cannot be applied directly. ...
We present a new approach to optimal design with state constraints based on active set optimization theory and implement it using a phase-field model. Our primary focus is on compliance minimization subject to inner and outer obstacles. We compare our approach to a classical penalization method and study the influence of initial guess, penalization parameters, and discretization.
... This specific problem has received much more attention since it is a more tractable case. See for instance [41,57]. ...
Multi-Leader-Follower games are complex optimization problems that mix a bilevel structure with one or more Nash games. Such kinds of models have been already described in the seminal book of H. von Stackelberg ((1934)Marktform und Gleichgewicht. Springer, Berlin); von Stackelberg et al. ((2011) Market structure and equilibrium. Springer, Heidelberg) and are known to perfectly fit to a lot of applications involving non cooperative situations with hierarchical interactions. Nevertheless it is only recently that theoretical and numerical developments for Multi-Leader-Follower problems have been made. This chapter aims to propose a state of the art of this field of research at the frontier between optimization and economics.
... This specic problem has received much more attention since it is a more tractable case. See for instance [92,121]. ...
This thesis is within the framework of optimization and deals with nonsmooth optimization and with some problems of game theory. It is divided into four parts. In the first introductory part, we give the context and some preliminary results. In the second part we discuss about subdifferential calculus rules in general spaces providing of some improved formulas in both the convex and the non-convex cases. Here the focus is on approximate or fuzzy calculus rules and optimality conditions, for which no qualification conditions are required. In the third part, we discuss about the so-called Multi-Leader-Follower Games. We give an existence result for the case of a single optimistic leader and multiple followers, and extend some results concerning the relation between the original problem with the reformulation obtained by replacing the followers' problem by the concatenation of their KKT conditions. Finally, in the fourth part we study quasi-equilibrium problems which are a general formulation for studying Nash equilibrium problems and quasi-variational inequalities. We provide some new existence results that relax some of the standard hypotheses.
... When the solution set of the variational inequality (3.2) is singleton, the MPEC problem (3.1) can be restated as an optimization problem in the variable y. This optimization problem was solved by numerical methods based on nonsmooth optimization techniques [77,78], gradient-based methods [11]. If, in addition, Y is a box, then the problem in the variable y was solved by sensitivity analysis based heuristic algorithms [20]. ...
In this dissertation, we investigate approaches based on DC (Difference of Convex functions) programming and DCA (DC Algorithm) for mathematical programs with equilibrium constraints. Being a classical and challenging topic of nonconvex optimization, and because of its many important applications, mathematical programming with equilibrium constraints has attracted the attention of many researchers since many years. The dissertation consists of four main chapters. Chapter 2 studies a class of mathematical programs with linear complementarity constraints. By using four penalty functions, we reformulate the considered problem as standard DC programs, i.e. minimizing a DC function on a convex set. The appropriate DCA schemes are developed to solve these four DC programs. Two among them are reformulated again as general DC programs (i.e. minimizing a DC function under DC constraints) in order that the convex subproblems in DCA are easier to solve. After designing DCA for the considered problem, we show how to develop these DCA schemes for solving the quadratic problem with linear complementarity constraints and the asymmetric eigenvalue complementarity problem. Chapter 3 addresses a class of mathematical programs with variational inequality constraints. We use a penalty technique to recast the considered problem as a DC program. A variant of DCA and its accelerated version are proposed to solve this DC program. As an application, we tackle the second-best toll pricing problem with fixed demands. Chapter 4 focuses on a class of bilevel optimization problems with binary upper level variables. By using an exact penalty function, we express the bilevel problem as a standard DC program for which an efficient DCA scheme is developed. We apply the proposed algorithm to solve a maximum flow network interdiction problem. In chapter 5, we are interested in the continuous equilibrium network design problem. It was formulated as a Mathematical Program with Complementarity Constraints (MPCC). We reformulate this MPCC problem as a general DC program and then propose a suitable DCA scheme for the resulting problem
... This problem belongs to a wider class of problems called mathematical programs with complementarity constraints (MPCC), which, in turn, constitute a subclass of mathematical programs with equilibrium constraints (MPEC). MPECs have been subject to extensive studies in the recent past both in functional spaces [5,7,8,9,10,33,45,41,46,48,49] and in finite dimensions [54,56,24]. ...
In the recent past the adaptive finite element method has proved to be successfully applicable for the efficient numerical solution of a large number of problems. The main focus of this thesis lies in the development of an adaptive finite element scheme based on a standard residual-type a posteriori error estimate for the numerical solution of distributed optimal control problems for second order elliptic variational inequalities of obstacle type.
As one of the main results, a convergence result is proven for a sequence of discrete C-stationary points. Furthermore, the residual-type a posteriori error estimator is shown to be reliable and efficient. Particular emphasis is put on the approximation of the reliability and efficiency related consistency errors. A detailed documentation of numerical results for a selection of test examples illustrates the performance of the adaptive approach.
... In particular, [40] describes some iterative algorithms, such as the penalty interior point algorithm (PIPA) and the piecewise sequential quadratic programming (PSQP) algorithm. Other early contributions are contained in [20,21,22,44,45]. More recent advances in MPEC algorithms can be found, for example, in [11,14,24,25,26,35,47]. ...
... Among the techniques proposed over the years to tackle an MPEC include branch and bound algorithms, implicit programming, exact penalisation, etc. (Jiang and Ralph, 2000;Fukushima et al., 1998;Zhang and Liu, 2001;Liu and Zhang, 2002;Scholtes and Stöhr, 1999;Outrata and Zowe, 1995;Dempe and Bard, 2001). ...
... Korpelevich (1977) introduces a two-step method for solving certain equilibrium problems and Miller et al. (1975) and Payne and Thompson (1975) introduce the "queue equilibrium" conditionj both ideas are used here. Also Outrata and Zowe (1995) have used a two-step method to solve "design" problems of a similar nature to the one being considered here. Gauvin and Savard (1994) have pursued a similar approach to solving bi-level problems. ...
There is an increasing determination to reduce congestion and other problems associated with the use of the motor car in cities; partly by encouraging mode shifts. For example, specific targets for increasing the use of public transport by 2020 have been suggested by the Royal Commission on Environmental Pollution in the United Kingdom (1994) and the Commission has very recently emphasised the lack of progress so far achieved.
... Concerning the mathematical theory of VI and QVI, the reader is referred to [2,4,8,10]. In the later works Outrata and Zowe [14] and O-utrata, Kocvara and Zowe [13] GNE is modeled via so-called mathematical program with equilibrium constraints (MPEC) and in the latter work a corresponding mathematical theory about MPECs is developed. ...
A special class of generalized Nash equilibrium problems is studied. Both variational and quasi-variational inequalities are used to derive some results concerning the structure of the sets of equilibria. These results are applied to the Cournot oligopoly problem.
... In recent years different theoretical approaches have been taken to establish MPECalgorithms to find solutions numerically, see for example the monograph [Luo et al. (1996)] for a penalty interior point method and a smoothing method, [Outrata and Zowe (1995)] for a non-smooth implicit function approach or [Scholtes and Stöhr (1999)] for an exact penalization approach. A good survey of algorithms before 1999 can be found in Stöhrs ...
... Other methods, with roots in traditional nonlinear methods include interior point (Byrd et al., 1999;Liu and Sun, 2004;Benson et al., 2006) and trust region (Scholtes and Stöhr, 1999;Colson et al., 2005b) algorithms. Implicit programming approaches (Outrata, 1994;Outrata and Zowe, 1995;Outrata et al., 1998) have also gotten some attention in the literature, but tend to require fairly strict assumptions on the problem (Luo et al., 1996). Unfortunately, the majority of the problems considered in the MPEC literature do not have integral variables and, thus, the solution methods are not applicable to the discrete problems we consider here. ...
... R p : The implicit function, S(x), can be rewritten as the solution set of an equivalent optimization problem, via a gap function for variational inequality problems. That is, S(x) := fy j y 2 arg min y g(x; y); y 2 Y (x)g: (4) The gap function proposed by Smith 6] for variational inequality problems de ned over polyhedral sets is g(y) := X i2I ( T(y) T (y ? z i )] + ) p (5) where y i ; i 2 I; are the extreme points of the polyhedral feasible set, ] + := maxf0; g, and p > 1. ...
In 1, 2], Smith and colleagues present an algorithm for solving the bilevel programming problem. We show that the points reached by the algorithm are not stationary points of bilevel programs in general. We further show that, with a minor modiication, this method can be expressed as an inexact penalty method for gap function-constrained bilevel programs.
... The most practiced methods are arguably the basic subgradient method and the penalty method. The subgradient method relies upon sensitivity analysis of the lower-level problem to produce an estimate of the subgradient; a formula for obtaining the subgradient of a parametric network equilibrium problem and the conditions under which it holds were provided in [12,13]. The authors in that reference actually suggest embedding the subgradient within a bundle algorithm so as to obtain much faster and more robust descent method; Bundle codes for nonconvex problems are not, however, readily available and are quite time consuming to implement. ...
We present a method for solving bilevel revenue management problem on networks. In this problem, a network manager seeks to set link prices optimally on a network so as to maximize revenue, where the users' response to the prices, their route choices, is given by a secondary, parametric optimization problem. This is an important special case of the general bilevel programming problem. We demonstrate that local solutions to this problem can be arbitrarily bad, and that local algorithms in many cases will stop at very poor solutions. Through a geometrical study of the non convex nature of the implicit objective, we develop a heuristic procedure that, in addition to being very easy to implement, performs remarkably well on this class of problems, often converging to the global optimum.
... It is closely related to bilevel programming problems (BLPP) and often called as a generalized BLPP. Some methods for solving MPEC or BLPP have been proposed such as extreme point algorithms [4,23,24], the complementarity pivot algorithm [12], heuristic descent algorithms [9,13,16] and implicit function algorithms [5,8,17,18]. Recently, a trust region algorithm [20] for solving nonlinear MPEC was proposed and its global convergence was given under some conditions. ...
In order to find a global solution for a quadratic program with linear complementarity constraints (QPLCC) more quickly than some existing methods, we consider to embed a local search method into a global search method. To say more specifically, in a branch-and-bound algorithm for solving QPLCC, when we find a new feasible solution to the problem, we utilize an extreme point algorithm to obtain a locally optimal solution which can provide a better bound and help us to trim more branches. So, the global algorithm can be accelerated. A preliminary numerical experiment was conducted which supports the new algorithm.
Optimization techniques have been widely used for image restoration tasks, as many imaging problems may be formulated as minimization ones with the recovered image as the target minimizer. Recently, novel optimization ideas also entered the scene in combination with machine learning approaches, to improve the reconstruction of images by optimally choosing different parameters/functions of interest in the models. This chapter provides a review of the latest developments concerning the latter, with special emphasis on bilevel optimization techniques and their use for learning local and nonlocal image restoration models in a supervised manner. Moreover, the use of related optimization ideas within the development of neural networks in imaging will be briefly discussed.
Mathematical programs with equilibrium constraints (MPECs) represent a class of hierarchical programs that allow for modeling problems in engineering, economics, finance, and statistics. While stochastic generalizations have been assuming increasing relevance, there is a pronounced absence of efficient first/zeroth-order schemes with non-asymptotic rate guarantees for resolving even deterministic variants of such problems. We consider a subclass of stochastic MPECs (SMPECs) where the parametrized lower-level equilibrium problem is given by a deterministic/stochastic variational inequality problem whose mapping is strongly monotone, uniformly in upper-level decisions. Under suitable assumptions, this paves the way for resolving the implicit problem with a Lipschitz continuous objective via a gradient-free zeroth-order method by leveraging a locally randomized spherical smoothing framework. Efficient algorithms for resolving the implicit problem allow for leveraging any convexity property possessed by the implicit problem, which in turn facilitates the computation of approximate global minimizers. In this setting, we present schemes for single-stage and two-stage stochastic MPECs when the upper-level problem is either convex or nonconvex in an implicit sense. (I). Single-stage SMPECs. In single-stage SMPECs, in convex regimes, our proposed inexact schemes are characterized by a complexity in upper-level projections, upper-level samples, and lower-level projections of \({\mathcal {O}}(\tfrac{1}{\epsilon ^2})\), \({\mathcal {O}}(\tfrac{1}{\epsilon ^2})\), and \({\mathcal {O}}(\tfrac{1}{\epsilon ^2}\ln (\tfrac{1}{\epsilon }))\), respectively. Analogous bounds for the nonconvex regime are \({\mathcal {O}}(\tfrac{1}{\epsilon })\), \({\mathcal {O}}(\tfrac{1}{\epsilon ^2})\), and \({\mathcal {O}}(\tfrac{1}{\epsilon ^3})\), respectively. (II). Two-stage SMPECs. In two-stage SMPECs, in convex regimes, our proposed inexact schemes have a complexity in upper-level projections, upper-level samples, and lower-level projections of \({\mathcal {O}}(\tfrac{1}{\epsilon ^2})\), \({\mathcal {O}}(\tfrac{1}{\epsilon ^2})\), and \({\mathcal {O}}(\tfrac{1}{\epsilon ^2}\ln (\tfrac{1}{\epsilon }))\) while the corresponding bounds in the nonconvex regime are \({\mathcal {O}}(\tfrac{1}{\epsilon })\), \({\mathcal {O}}(\tfrac{1}{\epsilon ^2})\), and \({\mathcal {O}}(\tfrac{1}{\epsilon ^2}\ln (\tfrac{1}{\epsilon }))\), respectively. In addition, we derive statements for accelerated schemes in settings where the exact solution of the lower-level problem is available. Preliminary numerics suggest that the schemes scale with problem size, are relatively robust to modification of algorithm parameters, show distinct benefits in obtaining near-global minimizers for convex implicit problems in contrast with competing solvers, and provide solutions of similar accuracy in a fraction of the time taken by sample-average approximation (SAA).
In this study, we discuss how the sequential quadratic Hamiltonian (SQH) scheme can be used to solve optimal control problems with non-smooth state constraints or non-smooth state cost terms. Theoretical and numerical aspects are both considered. Further, we connect to other recent techniques to solve such optimal control problems. The capability to handle non-smooth state terms extends the possibilities of modeling costs of the state and the dynamics of a system, e.g. to weight the deviation of the state from a desired state in an L1-norm. In this work, the non-smooth state terms are transformed into a bilinear structure on which the SQH scheme proved to solve optimal control problems fast promising an efficient numerical solution also in the non-smooth case. The novelty of the presented approach is that the non-smoothness is transformed into a bilinear structure. It is proved that the SQH method, which is based on the Pontryagin maximum principle, converges to an optimal solution of the corresponding transformed optimal control problem without any globalization techniques.
Optimization techniques have been widely used for image restoration tasks, as many imaging problems may be formulated as minimization ones with the recovered image as the target minimizer. Recently, novel optimization ideas also entered the scene in combination with machine learning approaches, to improve the reconstruction of images by optimally choosing different parameters/functions of interest in the models. This chapter provides a review of the latest developments concerning the latter, with special emphasis on bilevel optimization techniques and their use for learning local and nonlocal image restoration models in a supervised manner. Moreover, the use of related optimization ideas within the development of neural networks in imaging will be briefly discussed.
In this paper, we provide a finitely terminated yet efficient approach to compute the Euclidean projection onto the ordered weighted ℓ1 (OWL1) norm ball. In particular, an efficient semismooth Newton method is proposed for solving the dual of a reformulation of the original projection problem. Global and local quadratic convergence results, as well as the finite termination property, of the algorithm are proved. Numerical comparisons with the two best-known methods demonstrate the efficiency of our method. In addition, we derive the generalized Jacobian of the studied projector which, we believe, is crucial for the future designing of fast second-order nonsmooth methods for solving general OWL1 norm constrained problems.
The generalized Nash equilibrium problem is a kind of noncooperative game, whose strategy set of each player depends on the other rivals' decisions. Projection-type methods can be used to solve the corresponding quasi-variational inequalities problem. In this paper, we propose an accelerated method based on the half-space projection algorithm. At each iteration, we get a prediction point firstly by the half-space projection method. Then we use the extrapolation technique to obtain the next iterate. Numerical results show that our algorithm is effective.
In this paper, we provide a finitely terminated yet efficient approach to compute the Euclidean projection onto the ordered weighted $\ell_1$ (OWL1) norm ball. In particular, an efficient semismooth Newton method is proposed for solving the dual of a reformulation of the original projection problem. Global and local quadratic convergence results, as well as the finite termination property, of the algorithm are proved. Numerical comparisons with the two best-known methods demonstrate the efficiency of our method. In addition, we derive the generalized Jacobian of the studied projector which, we believe, is crucial for the future designing of fast second order nonsmooth methods for solving general OWL1 norm constrained problems.
In this paper, a feedback neural network model is proposed to compute the solution of the mathematical programs with equilibrium constraints (MPEC). The MPEC problem is altered into an identical one-level non-smooth optimization problem, then a sequential dynamic scheme that progressively approximates the non-smooth problem is presented. Besides asymptotic stability, it is proven that the limit equilibrium point of the suggested dynamic model is a solution for the original MPEC problem. Numerical simulation of various types of MPEC problems shows the significance of the results. Moreover, the scheme is applied to compute the Stackelberg-Cournot-Nash equilibria.
This chapter provides a short survey of the research for an important class of constrained optimization problems for which their constraints are defined in part by a variational inequality. Such problems are known as mathematical programs with equilibrium constraints (MPEC). MPEC arise naturally in different areas and play an important role, for example, in the pricing of telecommunication and transportation networks, in economic modeling, in computational mechanics in many other fields of modern optimization, and have been the subject of a number of recent studies. We present a general formulation of MPEC, describe the main characteristics of MPEC, and review the main properties and theoretical results for these problems. The short survey mainly concentrates on the review of the available solution methodology.
The paper deals with the application of a nonsmooth variant of the Newton method and a bundle method to the computation of Cournot-Nash, Walras and generalized (constrained) Nash equilibrium, respectively. In particular, conditions are examined ensuring the convergence.
本文给出了求解拟变分不等式问题的一种投影收缩算法，算法包括预测步和修正步，修正步的计算不需要做投影，在适当的假设条件下，证明了算法的全局收敛性。 In this paper, we present a projection and contraction algorithm for solving the quasi-variational inequality problem. The algorithm includes prediction step and correction step. The calculation of the correction step does not need to do the projection. Our method is proven to be globally con-vergent under certain assumptions.
The generalized quasi-variational inequality is a generalization of the generalized variational inequality and the quasi-variational inequality. The study for the generalized quasi-variational inequality is mainly concerned with the solution existence theory. In this paper, we present a cutting hyperplane projection method for solving generalized quasi-variational inequalities. Our method is new even if it reduces to solve the generalized variational inequalities. The global convergence is proved under certain assumptions. Numerical experiments have shown that our method has less total number of iterative steps than the most recent projection-like methods of Zhang et al. (Comput Optim Appl 45:89–109, 2010) for solving quasi-variational inequality problems and outperforms the method of Li and He (J Comput Appl Math 228:212–218, 2009) for solving generalized variational inequality problems.
A neural network approach is proposed for the treatment of inverse problems in linear and nonlinear elastostatics. The specific application concerns the identification of a crack in an elastic body with possible unilateral contact effects. The method is tested with numerical data produced by the multiregion boundary element method.
We study the behavior of a sequence generated by a smoothing continuation method for mathematical programs with equilibrium constraints (MPEC). In particular, we show that under the linear independence constraint qualification and an additional condition called the asymptotic weak nondegeneracy, the limit of KKT points satisfying the second-order necessary conditions for the perturbed problems is a B-stationary point of the original MPEC. The notable fact is that this result does not rely on the strict complementarity assumption at the limit point.
We are concerned with the numerical solution of distributed optimal control problems for second order elliptic variational inequalities by adaptive finite element methods. Both the continuous problem as well as its finite element approximations represent subclasses of Mathematical Programs with Equilibrium Constraints (MPECs) for which the optimality conditions are stated by means of stationarity concepts in function space (Hintermüller and Kopacka, SIAM J. Optim. 20:868–902, 2009) and in a discrete, finite dimensional setting (Scheel and Scholtes, Math. Oper. Res. 25:1–22, 2000) such as ("-almost, almost) C- and S-stationarity. With regard to adaptive mesh refinement, in contrast to the work in (Hintermüller, ESAIM Control Optim. Calc. Var., 2012, submitted) which adopts a goal oriented dual weighted approach, we consider standard residual-type a posteriori error estimators.The first main result states that for a sequence of discrete C-stationary points there exists a subsequence converging to an almost C-stationary point, provided the associated sequence of nested finite element spaces is limit dense in its continuous counterpart. As the second main result, we prove the reliability and efficiency of the residual-type a posteriori error estimators. Particular emphasis is put on the approximation of the reliability and efficiency related consistency errors by heuristically motivated computable quantities and on the approximation of the continuous active, strongly active, and inactive sets by their discrete counterparts.A detailed documentation of numerical results for two representative test examples illustrates the performance of the adaptive approach.
Keywords Some Examples Combinatorics Divisor Complement Ball Quotients Logarithmic Forms Hypergeometric Integrals See also References
本文给出了求解拟变分不等式问题的一种投影算法，在算法的第二次投影步中，把到一般闭凸集上的投影松弛为到半空间的投影，这在一定程度上减少了计算的难度。该算法的全局收敛性得到证明。 In this paper, we present a projection-like algorithm for solving the quasi-variational inequality problem. In the second projection step of the algorithm, we replace the orthogonal projection onto a general closed convex set with a projection onto a halfspace, which reduces the difficulty of cal-culation to some extent. The global convergence of the algorithm is given.
In this paper, we proposed an interior point method for constrained
optimization, which is characterized by the using of quasi-tangential
subproblem. This algorithm follows the main ideas of primal dual interior point
methods and Byrd-Omojokun's step decomposition strategy. The quasi-tangential
subproblem is obtained by penalizing the null space constraint in the
tangential subproblem. The resulted quasi-tangential step is not strictly lying
in the null space of the gradients of constraints. We also use a line search
trust-funnel-like strategy, instead of penalty function or filter technology,
to globalize the method. Global convergence results were obtained under
standard assumptions.
This paper addresses the systematic synthesis of contact-aided compliant mechanisms which use intermittent contact interactions to enable nonsmooth responses with a single-piece elastic body. The intermittent contact, assumed to be frictionless and adhesion-less in this paper, with a set of rigid external surfaces helps satisfy prescribed nonsmooth function or path generation requirements. The nondifferentiability inherent in the unilateral displacement constraints used to enforce contact is circumvented by introducing a regularized contact model. A small-strain, large rotation, updated Lagrangian FE formulation is used in conjunction with potential surface representation of external contact surfaces to accurately model the large-deformation behavior of a path-generating compliant mechanism. Element-level instabilities that arise during the intermediate stages of the optimization are addressed by a simple heuristic scheme. A numerical example is used to demonstrate the synthesis procedure. The experience with this approach suggests that an alternative approach based on the use of an arc-length static FE solver and a Fourier descriptor-based objective would be more appropriate for some problems. Copyright © 2004 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.
This paper discusses a kind of mathematical programs with equilibrium constraints (MPEC for short). By using a complementarity function and a kind of disturbed technique, the original (MPEC) problem is transformed into a nonlinear equality and inequality constrained optimization problem. Then, we combine a generalized gradient projection matrix with penalty function technique to given a generalized project metric algorithm with arbitrary initial point for the (MPEC) problems. In order to avoid Mataros effect, a high-order revised direction is obtained by an explicit formula. Under some relative weaker conditions, the proposed method is proved to possess global convergence and superlinear convergence.
A mathematical program with complementarity constraints (MPCC) is reformulated as a nonsmooth constrained mathematical program via the Fischer-Burmeister function. Quadratic penalty functions are used to treat this nonsmooth constrained program. We investigate necessary and sufficient conditions that guarantee the convergence of optimal values of unconstrained penalized problems to the optimal value of the original MPCC.
Optimal structural design problems concern with the choice of the design parameters (material variables, shapes etc) of a structure such as to make optimal certain response variables; this task is measured by an appropriate cost or gain function. They are mathematical programs with equilibrium constraints. The use of bilevel optimization techniques permit us to separate the optimal design problem and the structural analysis problem in two levels (subproblems) and, accordingly, allow for the use of existing methods and software for the solution of each one subproblem. In particular the utilisation of the powerful finite element analysis software is indispensable, since it permits the effective treatment of large scale problems.
Here the optimal structural design problem for nonsmooth mechanics’ structures is studied. All structures which involve non-differentiable potential energy functions and / or inequalities in their definition lead to nonsmooth mechanics problems. They are highly nonlinear problems which include, among others, frictionless and frictional unilateral contact effects, elastoplasticity etc. The structural analysis problem leads to nonsmooth potential energy minimization problems (or to substationarity problems in the case of nonconvex potentials). The nondifferentiability and the possible nonconvexity at the structural analysis level must be taken into account for the solution of the optimal design problem. In this chapter bilevel optimization algorithms for the solution of this problem will be investigated. Two model problems for which appropriate modelling techniques have been formulated and studied are presented here. They are solved by using previously published results and bilevel algorithms based on a coupling of nonsmooth optimization techniques with finite element modelling. The general problem is open for further research, a fact which is hoped to stimulate people working on multilevel and global optimization to contribute in this area.
In the paper The solution behaviour of a class of parameter-dependent quasi-variational Inequalities is analysed. By using sensitivity and stability results for monotone variational inequalities and the Implicit Function Theorem of F. H. Clarke, we derive conditions under which the perturbed solution of a parametric quasi-variational inequality is locally unique, lipschitzian and directionally differentiable. These results are particularized in the case of parametric implicit complementary problems.
Motivated by network design problems, we study a special class of mathematical programs with equilibrium constraints to which the so-called implicit programming approach cannot be applied directly. By using the tools of nonsmooth analysis, however, also these problems may be converted to Lipschitz programs, solvable by existing methods of nondifferentiable optimization.
The paper is devoted to optimization problems, where a quasivariational inequality arises as a side constraint. We augment this constraint partially to the objective and show that under a regularity condition the applied penalty is exact.
The contribution concerns mathematical programs in which a mixed complementarity problem arises as a side constraint. The attention is focussed above all to optimality conditions and to the respective constraint qualifications. In addition, an exact penalty approach to the numerical solution of such problems is proposed.
The paper concerns a two-level hierarchical game, where the players on each level behave noncooperatively. In this way one can model e.g., an oligopolistic market with several large and several small firms. We derive two types of necessary conditions for a solution of this game and discuss briefly the possibilities of its computation.
We describe some first- and second-order optimality conditions for mathematical programs with equilibrium constraints (MPEC). Mathematical programs with parametric nonlinear complementarity constraints are the focus. Of interest is the result that under a linear independence assumption that is standard in nonlinear programming, the otherwise combinatorial problem of checking whether a point is stationary for an MPEC is reduced to checking stationarity of single nonlinear program. We also present a piecewise sequential quadratic programming (PSQP) algorithm for solving MPEC. Local quadratic convergence is shown under the linear independence assumption and a second-order sufficient condition. Some computational results are given.
We propose a smoothing approach based on neural network function to solve a mathematical programme with equilibrium constraints (MPEC) in which the constraints are defined by a parametric variational inequality (PVI). We reformulate MPEC as an equivalent one level non-smooth optimisation problem. Then, this non-smooth optimisation problem will transfer to a sequence of smooth optimisation problems that can be solved by standard available software for constrained optimisation. Our results obtained in this paper continue to hold for any mathematical programme with parametric nonlinear complementarity/mixed complementarity constraints. Also, we test the performance of the proposed smoothing approach on a set of well-known problems and give some comparisons between our approach and other smoothing approaches. We are hoping our smoothing approach via neural network function will provide a basis for future applications work in this area, and generate some dynamic interactions between algorithmic developers and modellers/practitioners in the MPEC field.
The equilibrium network design problem can be formulated as a mathematical program with variational inequality constraints. We know this problem is nonconvex; hence, it is difficult to solve for a globally optimal solution. In this paper we propose a simulated annealing algorithm for the equilibrium network design problem. We demonstrate the ability of this algorithm to determine a globally optimal solution for two different networks. One of these describes an actual city in the midwestern United States.
NLPQL is a FORTRAN implementation of a sequential quadratic programming method for solving nonlinearly constrained optimization
problems with differentiable objective and constraint functions. At each iteration, the search direction is the solution of
a quadratic programming subproblem. This paper discusses the organization of NLPQL, including the formulation of the subproblem
and the information that must be provided by a user. A summary is given of the performance of different algorithmic options
of NLPQL on a collection of test problems (115 hand-selected or application problems, 320 randomly generated problems). The
performance of NLPQL is compared with that of some other available codes.
A generalized Nash game is an n-person noncooperative game with nondisjoint strategy sets; other names for this game form include social equilibria and pseudo-Nash games. This paper explores both the qualitative and quantitative properties of such games through the use of quasi-variational inequality theory. Several interesting relationships between the variational and quasi-variational inequality forms of this class of games are described and the practical implementation of generalized Nash games are explored at length.
Over the past decade, the field of finite-dimensional variational inequality and complementarity problems has seen a rapid development in its theory of existence, uniqueness and sensitivity of solution(s), in the theory of algorithms, and in the application of these techniques to transportation planning, regional science, socio-economic analysis, energy modeling, and game theory. This paper provides a state-of-the-art review of these developments as well as a summary of some open research topics in this growing field.
We prove an implicit function theorem for a class of generalized equations defined by a monotone set-valued mapping in Hilbert spaces. We give applications to variational inequalities, single-valued functions and a class of nonsmooth functions.
An algorithm (first outlined in the IX Symposium on Mathematical Programming) for the optimization of functions whose gradient is not continuous is presented. Motivation is emphasized. The algorithm is given in detail, including aspects of implementation, and its convergence is proved. It is also shown how this algorithm can handle simple linear constraints through an active set strategy.
In this paper, we introduce the generalized quasi-variational inequality problem and develop a theory for the existence of solution. This new problem includes as special cases two existing generalizations of the classical variational inequality problem. Relationship with a certain implicit complementarity problem is also studied.
During recent years various proposals for the minimization of a nonsmooth functional have been made. Amongst these, the bundle concept turned out to be an especially fruitful idea. Based on this concept, a number of authors have developed codes that can successfully deal with nonsmooth problems. The aim of the paper is to show that, by adding some features of the trust region philosophy to the bundle concept, the end result is a distinguished member of the bundle family with a more stable behaviour than some other bundle versions. The reliability and efficiency of this code is demonstrated on the standard academic test examples and on some real-life problems.
This paper presents a globally convergent model algorithm for the minimization of a locally Lipschitzian function. The algorithm is built on an iteration function of two arguments, and the convergence theory is developed parallel to analogous results for the problem of solving systems of locally Lipschitzian equations. Application of the theory to a wide range of nonsmooth optimization problems is discussed. These include the minimax problem, the composite optimization problem, the implicit programming problem, and others. A recently developed nonmonotone linesearch technique is shown to be applicable in this nonsmooth context, and an extension to constrained problems is also presented.
In this paper, we establish two characterization theorems for the linear and non-linear complementarity problems. Theorem 1 concerns the local uniqueness of a solution to a linear complementarity problem. Theorem 2 provides a necessary and sufficient condition for the differentiability of a solution to a parametric non-linear complementarity problem.
In this paper we introduce the concept of strong approximation of functions, and a related concept of strong Bouligand (B-) derivative, and we employ these ideas to prove an implicit-function theorem for nonsmooth functions. This theorem provides the same kinds of information as does the classical implicit-function theorem, but with the classical hypothesis of strong Fréchet differentiability replaced by strong approximation, and with Lipschitz continuity replacing Fréchet differentiability of the implicit function. Therefore it is applicable to a considerably wider class of functions than is the classical theorem.
In the last part of the paper we apply this implicit function result to analyze local solvability and stability of perturbed generalized equations.
This paper considers generalized equations, which are convenient tools for formulating problems in complementarity and in mathematical programming, as well as variational inequalities. We introduce a regularity condition for such problems and, with its help, prove existence, uniqueness and Lipschitz continuity of solutions to generalized equations with parametric data. Applications to nonlinear programming and to other areas are discussed, and for important classes of such applications the regularity condition given here is shown to be in a certain sense the weakest possible condition under which the stated properties will hold
This paper uses recently developed theory of sensitivity analysis to explore the reaction of a Cournot-Nash equilibrium to a Stackelberg firm, and to analyze the effect of this reaction on the uniqueness of the Stackelberg-Cournot-Nash equilibrium. Some of the results presented are not new, but the methods used provide simpler proofs and a different perspective. More importantly, the methods used here allow the development of new conditions for a unique Stackelberg-Cournot-Nash equilibrium that extends those previously known. The methods used also provide for the development of an efficient algorithm for finding the equilibrium.
A penalty function method approach for solving a constrained bilevel optimization problem is proposed. In the algorithm, both the upper level and the lower level problems are approximated by minimization problems of augmented objective functions. A convergence theorem is presented. The method is applicable to the non-singleton lower-level reaction set case. Constraint qualifications which imply the assumptions of the general convergence theorem are given.
First-order necessary optimality conditions are derived for a class of two-level Stackelberg problems in which the followers'' lower-level problems are convex programs with unique solutions. To this purpose, generalized Jacobians of the marginal maps corresponding to followers'' problems are estimated. As illustrative examples, two discretized optimum design problems with elliptic variational inequalities are investigated. The theoretical results may be used also for the numerical solution of the Stackelberg problems considered by nondifferentiable optimization methods.
Recently much attention has been focused on multilevel programming, a branch of mathematical programming that can be viewed either as a generalization of min-max problems or as a particular class of Stackelberg games with continuous variables. The network design problem with continuous decision variables representing link capacities can be cast into such a framework. We first give a formal description of the problem and then develop various suboptimal procedures to solve it. Worst-case behaviour results concerning the heuristics, as well as numerical results on a small network, are presented.
We propose and analyze a primal-dual, infinitesimal method for locating Nash equilibria of constrained, non-cooperative games. The main object is a family of nonstandard Lagrangian functions, one for each player. With respect to these functions the algorithm yields separately, in differential form, directions of steepest-descent in all decision variables and steepest-ascent in all multipliers. For convergence we need marginal costs to be monotone and constraints to be convex inequalities. The method is largely decomposed and amenable for parallel computing. Other noteworthy features are: non-smooth data can be accommodated; no projection or optimization is needed as subroutines; multipliers converge monotonically upward; and, finally, the implementation amounts, in essence, only to numerical integration.
During the past several years it has become increasingly common to use mathematical programming methods for deriving economic equilibria of supply and demand. Well-defined approaches exist for the case of a single firm (monopoly) and for the case of many firms (perfect competition). In this paper a certain family of convex programs is formulated to determine equilibria for the case of a few firms (oligopoly). Solutions to this family of convex programs are shown to be Nash equilibria in the formal sense ofN person games. This equivalence leads to a mathematical programming-based algorithm for determining an oligopolistic market equilibrium.
This study tries to develop two new approaches to the numerical solution of Stackelberg problems. In both of them the tools of nonsmooth analysis are extensively exploited; in particular we utilize some results concerning the differentiability of marginal functions and some stability results concerning the solutions of convex programs. The approaches are illustrated by simple examples and an optimum design problem with an elliptic variational inequality.Diese Arbeit zielt auf eine Entwicklung von neuen Verfahren fr die numerische Lsung der Stackelbergproblemen. In beiden vorgeschlagenen Verfahren ntzt man die Mittel der nichtglatten Analysis aus. Besonders handelt es sich um eine Charakterisierung der verallgemeinerten Gradienten von marginalen Funktionen und einige Stabilittsergebnisse, die die Lsungen von konvexen Programmen betreffen. Die Verfahren sind durch einfache Beispiele und ein Optimum Design Problem mit einer elliptischen Variationsungleichung illustriert.
Using a fixed point theorem of Browder, the basic existence theorem of Lemke in linear complementarity theory is generalized to the nonlinear case.
The paper provides a descent algorithm for solving certain monotone variational inequalities and shows how this algorithm may be used for solving certain monotone complementarity problems. Convergence is proved under natural monotonicity and smoothness conditions; neither symmetry nor strict monotonicity is required.
The paper deals with the numerical solution of a class of optimum design problems in which the controlled systems are described
by elliptic variational inequalities. The approach is based on the description of (discretized) system operators by means
of generalized Jacobians and the subsequent usage of nondifferentiable optimization methods. As an application, an important
shape design problems is solved.
In this paper we consider heuristic algorithms for a special case of the generalized bilevel mathematical programming problem in which one of the levels is represented as a variational inequality problem. Such problems arise in network design and economic planning. We obtain derivative information needed to implement these algorithms for such bilevel problems from the theory of sensitivity analysis for variational inequalities. We provide computational results for several numerical examples.
In this paper we study solution differentiability properties for variational inequalities. We characterize Fréchet differentiability of perturbed solutions to parametric variational inequality problems defined on polyhedral sets. Our result extends the recent result of Pang and it directly specializes to nonlinear complementarity problems, variational inequality problems defined on perturbed sets and to nonlinear programming problems.
Variational inequalities have often been used as a mathematical programming tool in modeling various equilibria in economics and transportation science. The behavior of such equilibrium solutions as a result of the changes in problem data is always of concern. In this paper, we present an approach for conducting sensitivity analysis of variational inequalities defined on polyhedral sets. We introduce the notion of differentiability of a point-to-set mapping and derive continuity and differentiability properties regarding the perturbed equilibrium solutions, even when the solution is not unique. As illustrated by several examples, the assumptions made in this paper are in a certain sense the weakest possible conditions under which the stated properties are valid. We also discuss applications to some equilibrium problems such as the traffic equilibrium problem.
Thesis--George Washington University. Abstract and vita inserted. Includes bibliographical references (leaves 104-106).
Bundle Trust methods: Fortran codes for nondifferentiable optimization. User's Guide
- H Schramm
H. Schramm, "Bundle Trust methods: Fortran codes for nondifferentiable optimization. User's Guide," DFG Research Report No. 269, University of Bayreuth (Bayreuth, 1991 ).
Bundle Trust methods: Fortran codes for nondifferentiable optimization
- H Schramm
- H. Schramm