No full-text available
To read the full-text of this research,
you can request a copy directly from the authors.
Mathematical Programs with Equilibrium Constraints
Abstract
This book provides a solid foundation and an extensive study for an important class of constrained optimization problems known as Mathematical Programs with Equilibrium Constraints (MPEC), which are extensions of bilevel optimization problems. The book begins with the description of many source problems arising from engineering and economics that are amenable to treatment by the MPEC methodology. Error bounds and parametric analysis are the main tools to establish a theory of exact penalisation, a set of MPEC constraint qualifications and the first-order and second-order optimality conditions. The book also describes several iterative algorithms such as a penalty-based interior point algorithm, an implicit programming algorithm and a piecewise sequential quadratic programming algorithm for MPECs. Results in the book are expected to have significant impacts in such disciplines as engineering design, economics and game equilibria, and transportation planning, within all of which MPEC has a central role to play in the modelling of many practical problems.
... Mathematical Programs with Equilibrium Constraints (MPEC) are NLPs that have a parametric variational inequality or optimization problem as an constraint [22]. Such constraints can be under suitable conditions replaced by equivalent complementarity conditions. ...
... If there exists an i such that G i (w) = 0, H i (w) = 0, the multiplier-based stationary concepts may not be strong enough to characterize local minimizers. The MPCC-tailored theory presented here has its origins in [22,24,29,30,43]. ...
... If a point w * satisfies the condition above, in the MPCC literature it is said that geometric Bouligand stationarity (geometric B-stationarity) holds [22,43]. For computational purposes, algebraic stationarity concepts are more useful. ...
This paper examines solution methods for mathematical programs with comple-mentarity constraints (MPCC) obtained from the time-discretization of optimal control problems (OCPs) subject to nonsmooth dynamical systems. The MPCC theory and stationarity concepts are reviewed and summarized. The focus is on relaxation-based methods for MPCCs, which solve a (finite) sequence of more regular nonlinear programs (NLP), where a regularization/homotopy parameter is driven to zero. Such methods perform reasonably well on currently available benchmarks. However, these results do not always generalize to MPCCs obtained from nonsmooth OCPs. To provide a more complete picture, this paper introduces a novel benchmark collection of such problems, which we call NOSBENCH. The problem set includes 603 different MPCCs and we split it into a few representative subsets to accelerate the testing. We compare different relaxation-based methods, NLP solvers, homotopy parameter update and relaxation parameter steering strategies. Moreover, we check whether the obtained stationary points allow first-order descent directions, which may be the case for some of the weaker MPCC stationarity concepts. In the best case, the Scholtes' relaxation [1] with IPOPT [2] as NLP solver manages to solve 73.8% of the problems. This highlights the need for further improvements in algorithms and software for MPCCs.
... Another driving force behind this work comes from disjunctive programming, in particular from the observation that constraints can be naturally formulated in a function-in-set format whereby sets are nonconvex yet simple (to project onto). The template (P) lends itself to capture this scenario, taking full advantage of as the indicator of a nonconvex set, comprising structures typical for, e.g., complementarity, switching, and vanishing constraints, see the classical monographs [40,46] and the more recent contributions [23,41]. ...
... i.e., cc is the standard complementarity set. Problem (P), thus, reduces to (MPCC) minimize ( ) subject to ( ) ∈ cc , a mathematical problem with complementarity constraints (MPCC), see the classical monographs [40,46]. Note that standard inequality and equality constraints can be added without any difficulty and are omitted here for brevity of presentation. ...
A broad class of optimization problems can be cast in composite form, that is, considering the minimization of the composition of a lower semicontinuous function with a differentiable mapping. This paper investigates the versatile template of composite optimization without any convexity assumptions. First- and second-order optimality conditions are discussed. We highlight the difficulties that stem from the lack of convexity when dealing with necessary conditions in a Lagrangian framework and when considering error bounds. Building upon these characterizations, a local convergence analysis is delineated for a recently developed augmented Lagrangian method, deriving rates of convergence in the fully nonconvex setting.
... Specifically, the first level identifies the optimal offer a given GenCo should submit to the Day-Ahead Electricity Market and the second level represents the Market Operator clearing problem (see, for instance, [6,20,21] and the references therein for a wider discussion). To cope with the challenge of solving the resulting non-convex problem, the original bi-level program is typically reformulated as a Mathematical Program with Equilibrium Constraints (MPEC) [22][23][24]. The subsequent MPEC can be further formulated as a Mixed-Integer Bi-Linear Programming (MI(B)LP) problem and solved using specialized procedures/algorithms. ...
... In (8), the set O S defines the Strategic Player price/quantity feasible offer, which might comprise, for instance, cap/floor levels, budget/risk constraints or logic relation between offers of different units that must be satisfied. Due to the linear and continuous nature of (1)-(4), problem (7)-(9) is commonly recast as an MPEC problem [23], following the procedure presented in Appendix 1. However, despite the recent advances studied in the technical literature in designing optimization-based techniques and algorithms to handle MPEC-related problems [30], efficiently solving (7)-(9), in particular in realistic, large-scale instances, is still challenging. ...
Efficiently devising optimal offers for Generation Companies (GenCos) in Day-Ahead Electricity Markets is a challenging task. Most solution procedures found in technical literature are built upon non-convex optimization structures, known as Mathematical Programming with Equilibrium Constraints (MPECs), that are difficult to scale to realistic-size instances. Therefore, the main objective of this work is to propose an efficient procedure to aid GenCos to devise optimal offering strategies in Day-Ahead Electricity Markets composed of a single bidding area. Supported by a set of technical results and strong duality theory, a tailored procedure that can be executed in polynomial-time in the number of firms is constructed with global-optimality guarantees. Numerical experiments are conducted, benchmarking the proposed approach against the standard MPEC-derived procedure typically found in the technical literature to solve the offering problem. We found that the proposed solution approach grows (roughly) linearly with the instance size and significantly overcomes (in the order of 20–25 times faster) its counterpart in the most demanding instances. Furthermore, the scalability of the MPEC-derived procedure is challenged even for medium-scale instances, whilst the proposed polynomial-time procedure was able to handle all instances in a reasonable computational time.
... Another driving force behind this work comes from disjunctive programming, in particular from the observation that constraints can be naturally formulated in a function-in-set format whereby sets are nonconvex yet simple (to project onto). The template (P) lends itself to capture this scenario, taking full advantage of as the indicator of a nonconvex set, comprising structures typical for, e.g., complementarity, switching, and vanishing constraints, see the classical monographs [40,46] and the more recent contributions [24,41]. ...
... i.e., cc is the standard complementarity set. Problem (P), thus, reduces to (MPCC) minimize ( ) subject to ( ) ∈ cc , a mathematical problem with complementarity constraints (MPCC), see the classical monographs [40,46]. Note that standard inequality and equality constraints can be added without any difficulty and are omitted here for brevity of presentation. ...
A broad class of optimization problems can be cast in composite form, that is, considering the minimization of the composition of a lower semicontinuous function with a differentiable mapping. This paper discusses the versatile template of composite optimization without any convexity assumptions. First- and second-order optimality conditions are discussed, advancing the variational analysis of compositions. We highlight the difficulties that stem from the lack of convexity when dealing with necessary conditions in a Lagrangian framework and when considering error bounds. Building upon these characterizations, a local convergence analysis is delineated for a recently developed augmented Lagrangian method, deriving rates of convergence in the fully nonconvex setting.
... The quantities κ ⋆ LP and κ ⋆ QP are some condition numbers associated with the optimization parameters. For instance, under the linear independence constraint qualification (Luo, Pang, and Ralph 1996), the inverse matrix in (6) is always full-rank; the alignment of some constraints, on the other hand, will result in larger values of κ ⋆ LP , since it will yield larger dual variables. In this section, we assume standard constraint qualifications, including Mangasarian-Fromovitz constraint qualification, constant rank constraint qualification, and strong coherent orientation condition (Luo, Pang, and Ralph 1996), such that both κ ⋆ LP and κ ⋆ QP are bounded. ...
... For instance, under the linear independence constraint qualification (Luo, Pang, and Ralph 1996), the inverse matrix in (6) is always full-rank; the alignment of some constraints, on the other hand, will result in larger values of κ ⋆ LP , since it will yield larger dual variables. In this section, we assume standard constraint qualifications, including Mangasarian-Fromovitz constraint qualification, constant rank constraint qualification, and strong coherent orientation condition (Luo, Pang, and Ralph 1996), such that both κ ⋆ LP and κ ⋆ QP are bounded. We also assume that Θ is compact and m 1 + m 2 ≥ n z , where n z is the dimension of the variables in (4); this assumption is easy to be relaxed with a slightly more complicated (but not necessarily more insightful) bound, thus we make the restriction to streamline the presentation. ...
We study the expressibility and learnability of solution functions of convex optimization and their multi-layer architectural extension. The main results are: (1) the class of solution functions of linear programming (LP) and quadratic programming (QP) is a universal approximant for the smooth model class or some restricted Sobolev space, and we characterize the rate-distortion, (2) the approximation power is investigated through a viewpoint of regression error, where information about the target function is provided in terms of data observations, (3) compositionality in the form of deep architecture with optimization as a layer is shown to reconstruct some basic functions used in numerical analysis without error, which implies that (4) a substantial reduction in rate-distortion can be achieved with a universal network architecture, and (5) we discuss the statistical bounds of empirical covering numbers for LP/QP, as well as a generic optimization problem (possibly nonconvex) by exploiting tame geometry. Our results provide the **first rigorous analysis of the approximation and learning-theoretic properties of solution functions** with implications for algorithmic design and performance guarantees.
... The authority's problem (13) is formulated in the form of mathematical programming with equilibrium constraints (MPEC) (see, e.g., [2,11,18]). This section is dedicated to illustrating the transformation of the challenging-to-solve MPEC formulation into a single Mixed-Integer programming problem. ...
... (1), an optimal solution to the latter can be characterized via standard optimality conditions. To this end, we choose for N θ a neural network (NN)-based representation of a hybrid system, ultimately leading to an OCP equivalent to (1) in the form of a mathematical program with complementarity constraints (MPCC) [49], whose classical Karush-Kuhn-Tucker (KKT) conditions [8, §5.1] will be necessary and sufficient to characterize an optimal solution. ...
We consider the problem of designing a machine learning-based model of an unknown dynamical system from a finite number of (state-input)-successor state data points, such that the model obtained is also suitable for optimal control design. We propose a specific neural network (NN) architecture that yields a hybrid system with piecewise-affine (PWA) dynamics that is differentiable with respect to the network's parameters, thereby enabling the use of derivative-based training procedures. We show that a careful choice of our NN's weights produces a hybrid system model with structural properties that are highly favourable when used as part of a finite horizon optimal control problem (OCP). Specifically, we show that optimal solutions with strong local optimality guarantees can be computed via nonlinear programming (NLP), in contrast to classical OCPs for general hybrid systems which typically require mixed-integer optimization. In addition to being well-suited for optimal control design, numerical simulations illustrate that our NN-based technique enjoys very similar performance to state-of-the-art system identification methodologies for hybrid systems and it is competitive on nonlinear benchmarks.
... Meanwhile, since QP is convex and if the Slater condition holds, the Karush-Kuhn-Tucker (KKT) condition is sufficient for optimality (Boyd and Vandenberghe 2004). Therefore, optimizations at the lower level can be replaced by the corresponding KKT conditions, known as the mathematical program with equilibrium constraints (MPEC) (Luo, Pang, and Ralph 1996). Therefore, if a linear parametric model is considered, it is possible to solve (12) exactly using optimization software. ...
Successful machine learning involves a complete pipeline of data, model, and downstream applications. Instead of treating them separately, there has been a prominent increase of attention within the constrained optimization (CO) and machine learning (ML) communities towards combining prediction and optimization models. The so-called end-to-end (E2E) learning captures the task-based objective for which they will be used for decision making. Although a large variety of E2E algorithms have been presented, it has not been fully investigated how to systematically address uncertainties involved in such models. Most of the existing work considers the uncertainties of ML in the input space and improves robustness through adversarial training. We extend this idea to E2E learning and prove that there is a robustness certification procedure by solving augmented integer programming. Furthermore, we show that neglecting the uncertainty of COs during training causes a new trigger for generalization errors. To include all these components, we propose a unified framework that covers the uncertainties emerging in both the input feature space of the ML models and the COs. The framework is described as a robust optimization problem and is practically solved via end-to-end adversarial training (E2E-AT). Finally, the performance of E2E-AT is evaluated by a real-world end-to-end power system operation problem, including load forecasting and sequential scheduling tasks.
... Different concepts of stationarity were introduced in the litterature and various techniques were used to establish optimality conditions. We only mention [5,9,14,18] and the references therein. From algorithmic point of view, methods such as interior points [13] or sequential quadratic programming [4] have proven to be effective in finite dimensions. ...
... From the above reformulation, it becomes evident that mathematical programs governed by generalized equations can be considered a significant subset of mathematical programs with equilibrium constraints (MPECs) [6]. These possess applications that extend to engineering design and economic modeling [7,8]. ...
In this paper, we consider a sparse program with symmetric cone constrained parameterized generalized equations (SPSCC). Such a problem is a symmetric cone analogue with vector optimization, and we aim to provide a smoothing framework for dealing with SPSCC that includes classical complementarity problems with the nonnegative cone, the semidefinite cone and the second-order cone. An effective approximation is given and we focus on solving the perturbation problem. The necessary optimality conditions, which are reformulated as a system of nonsmooth equations, and the second-order sufficient conditions are proposed. Under mild conditions, a smoothing Newton approach is used to solve these nonsmooth equations. Under second-order sufficient conditions, strong BD-regularity at a solution point can be satisfied. An inverse linear program is provided and discussed as an illustrative example, which verified the efficiency of the proposed algorithm.
... There have been proposed a number of approximation algorithms such as relaxation and smoothing algorithms, penalty function algorithms, interior point algorithms, implicit programming algorithms, active-set identification algorithms, constrained equation algorithms, and nonsmooth algorithms for solving MPCC. See, e.g., [9,10,12,13,17,18,20,26] and the references therein for more details about MPCC. Another approach in dealing with bilevel program is based on the lower-level optimal value function and some optimality conditions and constraint qualifications were derived through this way [7,30,31]. ...
This paper focuses on developing effective algorithms for solving bilevel program. The most popular approach is to replace the lower-level problem by its Karush-Kuhn-Tucker conditions to generate a mathematical program with complementarity constraints (MPCC). However, MPCC does not satisfy the Mangasarian-Fromovitz constraint qualification (MFCQ) at any feasible point. In this paper, inspired by a recent work using the lower-level Wolfe duality (WDP), we apply the lower-level Mond-Weir duality to present a new reformulation, called MDP, for bilevel program. It is shown that, under mild assumptions, they are equivalent in globally or locally optimal sense. An example is given to show that, different from MPCC, MDP may satisfy the MFCQ at its feasible points. Relations among MDP, WDP, and MPCC are investigated. Furthermore, in order to compare the new MDP approach with the MPCC and WDP approaches, we design a procedure to generate 150 tested problems randomly and comprehensive numerical experiments showed that MDP has evident advantages over MPCC and WDP in terms of feasibility to the original bilevel programs, success efficiency, and average CPU time.
... So a O( k ) local convergence rate is achieved with = (c 1 ∕(c 2 + c 1 )) 1 j . With the similar arguments as in the proof of Theorem 1, we obtain the rest of Theorem 2. ◻ It is worth noting that although the local R-linear convergence can be achieved in any cases under A2, when ∈ ( 1 2 , 1) , verification whether a training model and algorithm satisfies the stronger decrease condition is a challenging problem [23,28,40]. ...
Training deep neural networks (DNNs) is an important and challenging optimization problem in machine learning due to its non-convexity and non-separable structure. The alternating minimization (AM) approaches split the composition structure of DNNs and have drawn great interest in the deep learning and optimization communities. In this paper, we propose a unified framework for analyzing the convergence rate of AM-type network training methods. Our analysis is based on the non-monotone j-step sufficient decrease conditions and the Kurdyka-Łojasiewicz (KL) property, which relaxes the requirement of designing descent algorithms. We show the detailed local convergence rate if the KL exponent θ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta$$\end{document} varies in [0, 1). Moreover, the local R-linear convergence is discussed under a stronger j-step sufficient decrease condition.
The purpose of this paper is to study variational inequality problem over the solution set of multiple-set split monotone variational inclusion problem. We propose an iterative algorithm with inertial method for finding an approximate solution of this problem in real Hilbert spaces. Strong convergence of the sequence of iterates generated from the proposed method is obtained under some mild assumptions. The iterative scheme does not require prior knowledge of operator norm. Also we present some applications of our main result to solve the bilevel programming problem, the bilevel monotone variational inequalities, the split minimization problem, the multiple-set split feasibility problem and the multiple set split variational inequality problem.
Ride-sharing is one of the effective method to reduce car ownership, thus it may have a profound impact on the reserve capacity of road network. However, it’s unclear the relationships among users’ ridesharing behaviors, the reserve capacity of road network and travel demand pattern. To this end, this paper builds a ridesharing trip-assignment model which considers users’ ridesharing choice, destination choice and path choice, and further proposes a bi-level programming for reserve capacity of road network with ride-sharing. The bi-level programming is then converted into an equivalent single-layer optimization problem by a conventional relaxation scheme. Finally, numerical experiments are conducted to provide valuable insights and examine the effectiveness of the proposed model. The results show that subsidizing ridesharing drivers can improve almost as much reserve capacity of road network as expanding link capacity without ridesharing. However, retrofitting High-Occupancy Toll (HOT) lane has limited impact on improving reserve capacity of road network.
В статье представлена математическая модель планирования состояний электроэнергетической системы в среднесрочном периоде с учетом правил, действующих на рынках электроэнергии России. Необходимость такой модели обусловлена потребностью планирования электроэнергетических режимов с учётом интересов потребителей, включающих их надежное электроснабжение при минимизации затрат на покупку электроэнергии. При планировании режимов необходимо принимать во внимание возможные действия поставщиков, стремящихся максимизировать собственную прибыль при узловой ценообразовании на рынках. В представленной модели период планирования разбит на несколько временных интервалов, задача решается с учётом балансовых ограничений в узлах ЭЭС, ограничений по допустимым значениям генерации, перетоков и объемов энергоресурсов. Рыночное равновесие моделируется одновременно в нескольких интервалах с учётом многократности и длительности взаимодействия. Рассмотрены подходы поиска равновесного состояния в многоинтервальной задаче. Приводятся результаты численного моделирования на примере упрощенной схемы реальной ЭЭС.
In this paper, we propose and study a new modified Tseng inertial iterative technique for solving Bilevel quasimonotone Variational Inequality Problem (BVIP) in the framework of Hilbert spaces. In addition, we establish a strong convergence result of the proposed iterative technique under some mild assumptions, the proposed iterative technique does not need prior knowledge of Lipschitz’s constant of the quasimonotone operator. Finally, we present a numerical example of our proposed methods in comparison with some recent results in the literature, also, we apply our iterative technique to image deblurring and optimal control problems.
Theories of reading comprehension have widely predicted a role for syntactic skills, or the ability to understand and manipulate the structure of a sentence. Yet, these theories are based primarily on English, leaving open the question of whether this remains true across typologically different languages such as English versus Chinese. There are substantial differences in the sentence structures of Chinese versus English, making the comparison of the two particularly interesting. We conducted a meta-analysis contrasting the relation between syntactic skills and reading comprehension in first language readers of English versus Chinese. We test the influence of languages as well as the influence of the grade and the metrics on the magnitude of this relation. We identified 59 studies published between 1986 and 2021, generating 234 effect sizes involving 15,212 participants from kindergarten to high school and above. The magnitude of effects was remarkably similar for studies of English (r = .54) and Chinese (r = .54) readers, with similarities at key developmental points and syntactic tasks. There was also some evidence of modulation by grade levels and the nature of syntactic tasks. These findings confirm theory-based predictions of the importance of syntactic skills to reading comprehension. Extending these predictions, demonstrating these effects for both English and Chinese suggests a universal influence of syntactic skills on reading comprehension.
This paper focuses on developing effective algorithms for solving a bilevel program. The most popular approach is to replace the lower-level problem with its Karush-Kuhn-Tucker conditions to generate a mathematical program with complementarity constraints (MPCC). However, MPCC does not satisfy the Mangasarian-Fromovitz constraint qualification (MFCQ) at any feasible point. In this paper, inspired by a recent work using the lower-level Wolfe duality (WDP), we apply the lower-level Mond-Weir duality to present a new reformulation, called MDP, for bilevel program. It is shown that, under mild assumptions, they are equivalent in globally or locally optimal sense. An example is given to show that, different from MPCC, MDP may satisfy the MFCQ at its feasible points. Relations among MDP, WDP, and MPCC are investigated. On this basis, we extend the MDP reformulation to present another new reformulation (called eMDP), which has similar properties to MDP. Furthermore, to compare two new reformulations with the MPCC and WDP approaches, we design a procedure to generate 150 tested problems randomly and comprehensive numerical experiments show that MDP has quite evident advantages over MPCC and WDP in terms of feasibility to the original bilevel programs, success efficiency, and average CPU time, whereas eMDP is far superior to all other three reformulations.
History: Accepted by Pascal Van Hentenryck, Area Editor for Computational Modeling: Methods & Analysis.
Funding: This work was supported by the National Natural Science Foundation of China [Grants 12071280 and 11901380].
Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2023.0108 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2023.0108 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .
Electric vehicles (EVs), as popular transportation carriers and flexible electric loads, couple both the power distribution networks and transportation networks. The pricing schemes of fast charging stations significantly affects the EVs’ on-tirp charging behaviors and the operations of coupled power and transportation networks. This interdisciplinary paper proposes an optimal dynamic pricing method for fast charging stations to boost charging network operator’ (CNO’s) profits and avoid excessive charging costs of EVs. Time-varying traffic flow and charging demands are generated by a dynamic traffic assignment simulation, where a boundedly rational dynamic user equilibrium model is presented to capture the cost sensitivity of EVs and gasoline vehicle users. Additionally, a market regulator supervises the CNO’s pricing, balancing the interests of the CNO and users via a regulation constraint. To address the optimal dynamic pricing issue, we propose a three-level Stackelberg game involving the distribution network operator, CNO, and users. The existence of a game equilibrium is assured by the fixed-point theory. A Gauss-Seidel iteration algorithm with inertia weight is designed to solve the pricing problem. The numerical results have corroborated the effectiveness of the proposed method, illuminating the effects of bounded rationality and market regulation on the CNO’s profits and users’ travel costs.
The book offers a succinct overview of the technical components of blockchain networks, also known as distributed digital ledger networks. Written from an academic perspective, it surveys ongoing research challenges as well as existing literature. Several chapters illustrate how the mathematical tools of game theory and algorithmic mechanism design can be applied to the analysis, design, and improvement of blockchain network protocols. Using an engineering perspective, insights are provided into how the economic interests of different types of participants shape the behaviors of blockchain systems. Readers are thus provided with a paradigm for developing blockchain consensus protocols and distributed economic mechanisms that regulate the interactions of system participants, thus leading to desired cooperative behaviors in the form of system equilibria. This book will be a vital resource for students and scholars of this budding field.
Today artificial intelligence systems support efficient management in different fields of social activities. In particular, congestion control in modern networks seems to be impossible without proper mathematical models of traffic flow assignment. Thus, the network design problem can be referred to as Stackelberg game with independent lower-level drivers acting in a non-cooperative manner to minimize individual costs. In turn, upper-level decision-maker seeks to minimize overall travel time in the network by investing in its capacity. Hence, the decision-maker faces the challenge with a hierarchical structure which solution important due to its influence on the sustainable development of modern large cities. However, well-known that a bilevel programming problem is often strongly NP-hard, while the hierarchical optimization structure of a bilevel problem raises such difficulties as non-convexity and disconnectedness. The present paper is devoted to the swarm intelligence technique for capacity optimization of a transportation network. To this end, we develop the bilevel evolutionary algorithm based on swarm intelligence to cope with the continuous transportation network design problem. The findings of the paper give fresh insights to transport engineers and algorithm developers dealing with network design.
This paper is concerned with the distributed optimal control of a time-discrete Cahn–Hilliard–Navier–Stokes system with variable densities. It focuses on the double-obstacle potential which yields an optimal control problem for a variational inequality of fourth order and the Navier–Stokes equation. The existence of solutions to the primal system and of optimal controls is established. The Lipschitz continuity of the constraint mapping is derived and used to characterize the directional derivative of the constraint mapping via a system of variational inequalities and partial differential equations. Finally, strong stationarity conditions are presented following an approach from Mignot and Puel.
In this paper, we propose and study a Bilevel quasimonotone Variational Inequality Problem (BVIP) in the framework of Hilbert space. We introduce a new modified inertial iterative technique with self-adaptive step size for approximating a solution of the BVIP. In addition, we established a strong convergence result of the proposed iterative technique with adaptive step-size conditions without prior knowledge of Lips-chitz's constant of the cost operators as well as the strongly monotonicity coefficient under some standard mild assumptions. Finally, we provide some numerical experiments to demonstrate the efficiency of our proposed methods in comparison with some recently announced results in the literature.
In recent years, the number of electric vehicles (EVs) and fast charging stations has increased rapidly. Generally, self-interested charging station operators in urban transportation networks aim to maximize their profits by optimizing charging price strategies, while EV users aim to minimize their travel costs via routing and charging station choices. The non-cooperative interaction between charging station operators and EV users forms a Stackelberg game. To obtain the optimal pricing of fast charging stations, it is necessary to solve a mathematical program with complementarity constraints, which is non-convex and difficult to solve. In this study, a conic relaxation method is proposed to relax the original problem. If certain dual variables are non-negative, the proposed conic relaxation is exact. Furthermore, if users’ routing choices are more likely to be irrational, or perfectly rational and meanwhile almost all the EV users with the same origin-destination pair choose one same path, then the conic relaxation is more likely to be exact. Under other conditions, a penalty convex concave procedure is employed to obtain an optimal or near-optimal solution to the original problem based on the solution of the relaxed problem. A mathematical proof is provided, and numerical tests verify the efficacy of the method.
We propose a new approach to solving bilevel optimization problems, intermediate between solving full-system optimality conditions with a Newton-type approach, and treating the inner problem as an implicit function. The overall idea is to solve the full-system optimality conditions, but to precondition them to alternate between taking steps of simple conventional methods for the inner problem, the adjoint equation, and the outer problem. While the inner objective has to be smooth, the outer objective may be nonsmooth subject to a prox-contractivity condition. We prove linear convergence of the approach for combinations of gradient descent and forward-backward splitting with exact and inexact solution of the adjoint equation. We demonstrate good performance on learning the regularization parameter for anisotropic total variation image denoising, and the convolution kernel for image deconvolution.
In this paper, we propose several new constraint qualifications for mathematical programs with second-order cone complementarity constraints (SOCMPCC), named SOCMPCC-K-, strongly (S-), and Mordukhovich (M-) relaxed constant positive linear dependence condition (K-/S-/M-RCPLD). We show that K-/S-/M-RCPLD can ensure that a local minimizer of SOCMPCC is a K-/S-/M-stationary point, respectively. We further give some other constant rank-type constraint qualifications for SOCMPCC. These new constraint qualifications are strictly weaker than SOCMPCC linear independent constraint qualification and nondegenerate condition. Finally, we demonstrate the relationships among the existing SOCMPCC constraint qualifications.
In this paper, we investigate the use of bilevel optimization for model learning in variational imaging problems. Bilevel learning is an alternative approach to traditional deep learning methods, that leads to fully interpretable models. However, it requires a detailed analytical insight into the underlying mathematical model. We focus on the bilevel learning problem for total variation models with spatially- and patch-dependent parameters. Our study encompasses the directional differentiability of the solution mapping, the derivation of optimality conditions, and the characterization of the Bouligand subdifferential of the solution operator. We also propose a two-phase trust-region algorithm for solving the problem and present numerical tests using the CelebA dataset.
In this paper, a class of smooth penalty functions is proposed for constrained optimization problem. It is put forward based on \(L_p\), a smooth function of a class of exact penalty function \({\ell _p}~\left( {p \in (0,1]} \right) \). Based on the class of penalty functions, a penalty algorithm is presented. Under the very weak condition, a perturbation theorem is set up. The global convergence of the algorithm is derived. This result generalizes some existing conclusions. Finally, numerical experiments on two examples demonstrate the effectiveness and efficiency of our algorithm.
In massive multi-user multi-input–multi-output (MU-MIMO) downlink transmission systems, the total power consumption of the base station is mainly attributed to high-resolution digital-to-analog converters (DACs). Therefore, utilizing one-bit DACs can effectively reduce the system power consumption. This work investigates the one-bit DACs precoding problem in massive MU-MIMO systems. Typically, the one-bit DACs precoding problem based on the minimum mean square error criterion is an NP-hard problem, making it challenging to solve. By exploiting the structural properties of nonconvex constraints, the nonconvex optimization problem is transformed into an equivalent continuous-domain mathematical programming problem with equilibrium constraints. The projected gradient method is used for obtaining the quantized precoding signals, and the method is optimized by introducing a step-size adjustment coefficient. Simulation results demonstrate that the improved projected gradient method can effectively reduce the uncoded bit error rate and achieve a performance gain of approximately
$\text{2}\,\text{dB}$
with 16 quadrature amplitude modulation signaling. The simulations also demonstrate the better robustness of the suggested method with imperfect channel state information. Specifically, we evaluate the power consumption of the proposed algorithm to that of the zero-forcing precoder with different resolution DACs, illustrating its superior power efficiency. Furthermore, we demonstrate the effectiveness of the proposed algorithm through a theoretical analysis of the convergence, complexity, and exact property of the optimal solution.
Gradient-based Bi-Level Optimization (BLO) methods have been widely applied to handle modern learning tasks. However, most existing strategies are theoretically designed based on restrictive assumptions (e.g., convexity of the lower-level sub-problem), and computationally not applicable for high-dimensional tasks. Moreover, there are almost no gradient-based methods able to solve BLO in those challenging scenarios, such as BLO with functional constraints and pessimistic BLO. In this work, by reformulating BLO into approximated single-level problems, we provide a new algorithm, named Bi-level Value-Function-based Sequential Minimization (BVFSM), to address the above issues. Specifically, BVFSM constructs a series of value-function-based approximations, and thus avoids repeated calculations of recurrent gradient and Hessian inverse required by existing approaches, time-consuming especially for high-dimensional tasks. We also extend BVFSM to address BLO with additional functional constraints. More importantly, BVFSM can be used for the challenging pessimistic BLO, which has never been properly solved before. In theory, we prove the convergence of BVFSM on these types of BLO, in which the restrictive lower-level convexity assumption is discarded. To our best knowledge, this is the first gradient-based algorithm that can solve different kinds of BLO (e.g., optimistic, pessimistic, and with constraints) with solid convergence guarantees. Extensive experiments verify the theoretical investigations and demonstrate our superiority on various real-world applications.
The disjunctive system is a system involving a disjunctive set which is the union of finitely many polyhedral convex sets. In this paper, we introduce a notion of the relaxed constant positive linear dependence constraint qualification (RCPLD) for the disjunctive system. For a disjunctive system, our notion is weaker than the one we introduced for a more general system recently (J. Glob. Optim. 2020) and is still a constraint qualification. To obtain the local error bound for the disjunctive system, we introduce the piecewise RCPLD under which the error bound property holds if all inequality constraint functions are subdifferentially regular and the rest of the constraint functions are smooth. We then specialize our results to the ortho-disjunctive program, which includes the mathematical program with equilibrium constraints (MPEC), the mathematical program with vanishing constraints (MPVC) and the mathematical program with switching constraints (MPSC) as special cases. For MPEC, we recover MPEC-RCPLD, an MPEC variant of RCPLD and propose the MPEC piecewise RCPLD to obtain the error bound property. For MPVC, we introduce new constraint qualifications MPVC-RCPLD and the piecewise RCPLD, which also implies the local error bound. For MPSC, we show that both RCPLD and the piecewise RCPLD coincide and hence it leads to the local error bound.
In this paper, we consider nonsmooth multiobjective optimization problems with equilibrium constraints. Necessary/sufficient conditions for optimality in terms of the Michel-Penot subdifferential are established. Then, we propose Wolfe- and Mond–Weir-types of dual problems and investigate duality relations under generalized convexity assumptions. Some examples are provided to illustrate our results.
In this paper, we consider nonsmooth multiobjective optimization problems with equilibrium constraints. Necessary/sufficient conditions for optimality in terms of the Michel-Penot subdifferential are established. Then, we propose Wolfe- and Mond–Weir-types of dual problems and investigate duality relations under generalized convexity assumptions. Some examples are provided to illustrate our results.
Problem definition: With the rise of renewables and the decline of fossil fuels, electricity markets are shifting toward a capacity mix in which low-cost generators (LCGs) are dominant. Within this transition, policymakers have been considering whether current market designs are still fundamentally fit for purpose. This research analyses a key aspect: the design of real-time imbalance pricing mechanisms. Currently, markets mostly use either single pricing or dual pricing as their imbalance pricing mechanisms. Single-pricing mechanisms apply identical prices for buying and selling, whereas dual-pricing mechanisms use different prices. The recent harmonization initiative in Europe sets single pricing as the default and dual pricing as the exception. This leaves open the question of when dual pricing is advantageous. We compare the economic efficiency of two dual-pricing mechanisms in current practice with that of a single-pricing design and identify conditions under which dual pricing can be beneficial. We also prove the existence of an optimal pricing mechanism. Methodology/results: We first analytically compare the economic efficiency of single-pricing and dual-pricing mechanisms. Furthermore, we formulate an optimal pricing mechanism that can deter the potential exercise of market power by LCGs. Our analytical results characterize the conditions under which a dual pricing is advantageous over a single pricing. We further compare the economic efficiency of these mechanisms with respect to our proposed optimal mechanism through simulations. We show that the proposed pricing mechanism would be the most efficient in comparison with others and discuss its practicability. Managerial implications: Our analytical comparison reveals market conditions under which each pricing mechanism is a better fit and whether there is a need for a redesign. In particular, our results suggest that existing pricing mechanisms are adequate at low/moderate market shares of LCGs but not for the high levels currently envisaged by policymakers in the transition to decarbonization, where the optimal pricing mechanism will become more attractive.
Funding: Rodrigo Belo acknowledges funding by Fundação para a Ciência e a Tecnologia (UIDB/00124/2020, UIDP/00124/2020 and Social Sciences DataLab – PINFRA/22209/2016), POR Lisboa, and POR Norte (Social Sciences DataLab, PINFRA/22209/2016).
Supplemental Material: The e-companion is available at https://doi.org/10.1287/msom.2021.0555 .
In this paper, we study an iterative algorithm that is based on inertial projection and contraction methods for solving bilevel quasimonotone variational inequality problems in the framework of real Hilbert spaces. We establish a strong convergence result of the proposed iterative method based on adaptive stepsizes conditions without prior knowledge of Lipschitz constant of the cost operator as well as the strongly monotonicity coefficient under some standard mild assumptions on the algorithm parameters. Finally, we present some special numerical experiments to show efficiency and comparative advantage of our algorithm to other related methods in the literature. The results presented in this article improve and generalize some well-known results in the literature.
The aim of this article is to study mathematical programs with equilibrium constraints involving quasidifferentiable functions, denoted by QMPEC, and to synthesize suitable optimality conditions. We first derive Fritz-John (FJ) necessary optimality conditions with Lagrange multipliers depending upon the choice of superdifferentials. We introduce a suitable variant of no nonzero abnormal multiplier constraint qualification for the QMPEC, denoted by NNAMCQ-QMPEC, and derive Karush–Kuhn–Tucker (KKT) necessary optimality conditions. We also propose some conditions under which the Lagrange multipliers do not depend upon the choice of superdifferentials. Further, we prove several sufficient optimality conditions for a weak stationary point to be optimal for the QMPEC under suitable choice of generalized convex functions.
ResearchGate has not been able to resolve any references for this publication.