Gert Wanka

Technische Universität Chemnitz · Department of Mathematics

The aim of this manuscript is to approach by means of first order differential equations/inclusions convex programming problems with two-block separable linear constraints and objectives, whereby (at least) one of the components of the latter is assumed to be strongly convex. Each block of the objective contains a further smooth convex function. We...

The aim of this manuscript is to approach by means of first order differential equations/inclusions convex programming problems with two-block separable linear constraints and objectives, whereby (at least) one of the components of the latter is assumed to be strongly convex. Each block of the objective contains a further smooth convex function. We...

The Alternating Minimization Algorithm has been proposed by Paul Tseng to solve convex programming problems with two-block separable linear constraints and objectives, whereby (at least) one of the components of the latter is assumed to be strongly convex. The fact that one of the subproblems to be solved within the iteration process of this method...

The Alternating Minimization Algorithm (AMA) has been proposed by Tseng to solve convex programming problems with two-block separable linear constraints and objectives, whereby (at least) one of the components of the latter is assumed to be strongly convex. The fact that one of the subproblems to be solved within the iteration process of AMA does n...

Duality statements are presented for multifacility location problems as suggested by Drezner Hiu 1991, where for each given point the sum of weighted distances to all facilities plus set-up costs is determined and the maximal value of these sums is to be minimized. We develop corresponding dual problems for the cases with and without set-up costs a...

In the framework of conjugate duality we discuss nonlinear and linear single minimax location problems with geometric constraints, where the gauges are defined by convex sets of a Fréchet space. The version of the nonlinear location problem is additionally considered with set-up costs. Associated dual problems for this kind of location problems wil...

In this paper, we consider an optimization problem with geometric and cone constraints, whose objective function is a composition of \(n+1\) functions. For this problem, we calculate its conjugate dual problem, where the functions involved in the objective function of the primal problem will be decomposed. Furthermore, we formulate generalized inte...

We deliver formulae for the biconjugate functions of some infimal functions that hold provided the fulfilment of weak regularity conditions of both closedness and interiority types. As special cases, we obtain biconjugates of infimal convolutions of finitely many functions, of optimal value functions of both constrained and unconstrained optimizati...

Supervised learning methods are powerful techniques to learn a function from a given set of labeled data, the so-called training data. In this paper the support vector machines approach is applied to an image classification task. Starting with the corresponding Tikhonov regularization problem, reformulated as a convex optimization problem, we intro...

This paper aims to extend some results dealing with gap functions for vector variational inequalities from the literature by using the so-called oriented distance function.

In this chapter we consider scalar and vector optimization problems with objective functions being the composition of a convex function and a linear mapping and cone and geometric constraints. By means of duality theory we derive dual problems and formulate weak, strong, and converse duality theorems for the scalar and vector optimization problems...

With this note we bring again into attention a vector dual problem neglected by the contributions who have recently announced the successful healing of the trouble encountered by the classical duals to the classical linear vector optimization problem. This vector dual problem has, different to the mentioned works which are of set-valued nature, a v...

This chapter deals with the so-called perturbation approach in the conjugate duality for vector optimization on the basis of weak orderings. As applications, we investigate some new set-valued gap functions for vector equilibrium problems. Keywordsconjugate duality-perturbation approach-vector equilibrium problems-set-valued gap functions

The aim of this paper is to give dual representations for different convex risk measures by employing their conjugate functions. To establish the formulas for the conjugates, we use on the one hand some classical results from convex analysis and on the other hand some tools from the conjugate duality theory. Some characterizations of so-called devi...

In this paper we deal with linear chance-constrained opti-mization problems, a class of problems which naturally arise in practical applications in finance, engineering, transportation and scheduling, where decisions are made in presence of uncertainty. After giving the determin-istic equivalent formulation of a linear chance-constrained optimizati...

We give two generalized Moreau-Rockafellar-type results for the sum of a convex function with a composition of convex functions in separated locally convex spaces. Then we equivalently characterize the stable strong duality for composed convex optimization problems through two new regularity conditions, which also guarantee two formulae of the subd...

The aim of this chapter is to describe the so-called conjugate duality theory for scalar optimization problems, which represents a cornerstone in the duality theory for vector optimization problems. The most duality concepts which can be found in the literature on vector optimization, this book being no exception, have the origin in well-developed...

Since the early eighties of the last century there have been attempts to extend the perturbation approach for scalar duality (as developed in chapter 3 from scalar optimization problems) to vector duality in connection with different generalizations of the conjugacy concept. The idea of conjugate functions and subdifferentiability in scalar optimiz...

In this chapter we introduce to different vector optimization problems corresponding vector dual problems by using some duality concepts having as starting point the scalar duality theory. Since there is a certain similarity between its definition and the one of the duals introduced via scalarization, we also investigate the classical geometric vec...

In this chapter we present scalar and vector duality based on the classical Wolfe and Mond-Weir duality concepts. As the field is very vast, especially because of different generalizations of the notion of convexity for the functions employed, we limited our exposition to a reasonable framework, large enough to present the most relevant facts in th...

In this chapter we introduce some basic notions and results in convex analysis and vector optimization in order to make the book as self-contained as possible. The reader is supposed to have basic notions of functional analysis.

In this chapter we introduce new conjugate vector dual problems to the primal problems treated in the previous chapter in case their objective functions have finite dimensional image spaces. Weak, strong and converse duality assertions are proven and these duals are compared with the ones introduced in chapter 4. Note that the properly efficient so...

We introduce a new Fenchel dual for vector optimization problems inspired by the form of the Fenchel dual attached to the scalarized primal multiobjective problem. For the vector primal-dual pair we prove weak and strong duality. Furthermore, we recall two other Fenchel-type dual problems introduced in the past in the literature, in the vector case...

In this paper we derive by means of the duality theory necessary and sufficient optimality conditions for convex optimization problems having as objective function the composition of a convex function with a linear mapping defined on a finite-dimensional space with values in a Hausdorff locally convex space. We use the general results for deriving...

We give new regularity conditions expressed via epigraphs that as- sure strong duality between a given primal convex optimization problem and its Lagrange and Fenchel-Lagrange dual problems, respectively, in innite dimen- sional spaces. Moreover we completely characterize through equivalent state- ments the so-called stable strong duality between t...

This book presents fundamentals and comprehensive results regarding duality for scalar, vector and set-valued optimization problems in a general setting. After a preliminary chapter dedicated to convex analysis and minimality notions of sets with respect to partial orderings induced by convex cones a chapter on scalar conjugate duality follows. The...

We deal with duality for almost convex finite dimensional optimization problems by means of the classical perturbation approach. To this aim some standard results from the convex analysis are extended to the case of almost convex sets and functions. The duality for some classes of primal-dual problems is derived as a special case of the general app...

When characterizing optimal solutions of both scalar and vector optimization problems usually constraint qualications have to be satised. By considering sequential characterizations, given for the rst time in vector optimization in this paper, this drawback is eliminated. In order to establish them we give rst of all sequential characterizations fo...

We consider a convex optimization problem with a vector valued function as objective function and convex cone inequality constraints. We suppose that each entry of the objective function is the composition of some convex functions. Our aim is to provide necessary and sufficient conditions for the weakly efficient solutions of this vector problem. M...

In this paper we use the tools of the convex analysis in order to give a suitable characterization for the epigraph of the conjugate of the pointwise maximum of two proper, convex and lower semicontinuous functions in a normed space. By using this characterization we obtain, as a natural consequence, the formula for the biconjugate of the pointwise...

In this paper we work in separated locally convex spaces where we give equivalent statements for the formulae of the conjugate function of the sum of a convex lower-semicontinuous function and the precomposition of another convex lower-semicontinuous function which is also K -increasing with a K -convex K -epi-closed function, where K is a nonempty...

The aim of the present paper is to provide a formula for the ɛ-subdifferential of f+g∘h different from the ones which can be found in the existent literature. Further we equivalently characterize this formula by using a so-called closedness type regularity condition expressed by means of the epigraphs of the conjugates of the functions involved. Ev...

We give new regularity conditions for convex optimization prob-lems in separated locally convex spaces. We completely characterize the stable strong and strong Fenchel-Lagrange duality. Then we give similar statements for the case when a solution of the primal problem is assumed as known, obtaining complete characterizations for the so-called total...

We introduce some abstract convexity notions in a real linear space and investigate which of the results from the convex analysis in topological vector spaces still work in a linear space. The differences between these abstract convexity notions and those established in spaces endowed with a topology are underlined by some examples.

Using a general approach which provides sequential optimality conditions for a general convex optimization problem, we derive necessary and sufficient optimality conditions for composed convex optimization problems. Further, we give sequential characterizations for a subgradient of the precomposition of a K-increasing lower semicontinuous convex fu...

We give some new regularity conditions for Fenchel duality in separated locally convex vector spaces, written in terms of the notion of quasi interior and quasi-relative interior, respectively. We provide also an example of a convex optimization problem for which the classical generalized interior-point conditions given so far in the literature can...

We give some necessary and sufficient conditions which completely characterize the strong and total Lagrange duality, respectively, for convex optimization problems in separated locally convex spaces. We also prove similar statements for the problems obtained by perturbing the objective functions of the primal problems by arbitrary linear functiona...

In this paper a necessary and sufficient sequential optimality condition without a constraint qualification for a general convex optimization problem is given in terms of the ε-subdifferential. Further, a sequential char-acterization of optimal solutions involving the convex subdifferential is derived using a version of the Brøndsted-Rockafellar Th...

We describe a supervised text classification approach based on a greedy feature selection method, which uses a support vector machine (SVM) classi-fier. As feature selection method we use the mutual information. This measures the quantity of information about the categories contained by the words. To train and test the algorithm we used patent docu...

We present some Farkas-type results for inequality systems involving finitely many DC functions. To this end we use the so-called Fenchel-Lagrange duality approach applied to an optimization problem with DC objective function and DC inequality constraints. Some recently obtained Farkas-type results are rediscovered as special cases of our main resu...

We present a new constraint qualification which guarantees strong duality between a cone-constrained convex optimization problem and its Fenchel-Lagrange dual. This result is applied to a convex optimization problem having, for a given nonempty convex cone K, as objective function a K-convex function postcomposed with a K-increasing convex function...

In this paper we give a weaker sufficient condition for the maximal monotonicity of the operator S + A * T A, where S : X X * , T : Y Y * are two maximal monotone operators, A : X → Y is a linear continuous mapping and X, Y are reflexive Banach spaces. We prove that our condition is weaker than the generalized interior-point conditions given so far...

Considering a constrained fractional programming problem, within the present paper we present some necessary and sufficient conditions, which ensure that the optimal objective value of the considered problem is greater than or equal to a given real constant. The desired results are obtained using the Fenchel–Lagrange duality approach applied to an...

The present paper is a continuation of [2] where we deal with the duality for a multiobjective fractional optimization problem. The basic idea in [2] consists in attaching an intermediate multiobjective convex optimization problem to the primal fractional problem, using an approach due to Dinkelbach ([6]), for which we construct then a dual problem...

Given a multiobjective optimization problem with the components of the objective function as well as the constraint functions being composed convex functions, we introduce, by using the Fenchel-Moreau conjugate of the functions involved, a suitable dual problem. Under a standard constraint qualification and some convexity as well as monotonicity co...

For an optimization problem with a composed objective function and composed constraint functions we determine, by means of the conjugacy approach based on the perturbation theory, some dual problems to it. The relations between the optimal objective values of these duals are studied. Moreover, sufficient conditions are given in order to achieve equ...

We present an extension of Fenchel’s duality theorem by weakening the convexity assumptions to near convexity. These weak hypotheses are automatically fulfilled in the convex case. Moreover, we show by a counterexample that a further extension to closely convex functions is not possible under these hypotheses.

In this paper, we deal with the construction of gap functions for variational inequalities by using an approach which bases on the conjugate duality. Under certain assumptions we also investigate a further class of gap functions for the variational inequality problem, the so-called dual gap functions.

Given an optimization problem with a composite of a convex and com- ponentwise increasing function with a convex vector function as objective function, by means of the conjugacy approach based on the perturbation theory, we determine a dual to it. Necessary and sucient optimality conditions are derived using strong duality. Furthermore, as special...

A general duality framework in convex multiobjective optimization is established using the scalarization with K-strongly increasing functions and the conjugate duality for composed convex cone-constrained optimization problems. Other scalarizations used in the literature arise as particular cases and the general duality is specialized for some of t...

The aim of this paper is to extend the so-called perturbation approach in order to deal with conjugate duality for constrained vector optimization problems. To this end we use two conjugacy notions introduced in the past in the literature in the framework of set-valued optimization. As a particular case we consider a vector variational inequality w...

We give a sufficient condition, weaker than the others known so far, that guarantees that the sum of two maximal monotone operators on a reflexive Banach space is maximal monotone. Then we give a weak constraint qualification assuring the Brézis-Haraux-type approximation of the range of the sum of the subdifferentials of two proper convex lower-sem...

We give the weakest constraint qualification known to us that ensures the maximal monotonicity of the operator A T A when A is a linear continuous mapping between two reflexive Banach spaces and T is a maximal monotone operator. As a special case we get the weakest constraint qualification that ensures the maximal monotonicity of the sum of two max...

We prove that the formulae of the conjugates of the precomposition with a linear operator, of the sum of finitely many functions and of the sum between a function and the precomposition of another one with a linear operator hold even when the convexity assumptions are replaced by almost convexity or nearly convexity. We also show that the duality s...

In this survey we present some of our recent results concerning regularity conditions for subdifferential calculus and Fenchel duality in infinite dimensional spaces. As an application we deliver the maximal monotonicity of the operator A * • T • A, where A is a linear continuous mapping between two re-flexive Banach spaces and T is a maximal monot...

In this paper we present first some Farkas-type results for inequality systems with convex and with composed convex functions, re-spectively, expressed by means of the conjugate functions of the functions involved. It is also shown that Motzkin's theorem of the alternative is ac-tually a special instance of the general result we give. Another appli...

This paper deals with the characterization of solutions for vector equilibrium problems by means of conjugate duality. By using the Fenchel duality we establish variational principles, that is, optimization problems with set-valued objective functions, the solution sets of which contain the ones of the vector equilibrium problems. The set-valued ob...

We give Brézis -Haraux -type approximation results for the range of the monotone operator S + A * • T • A when A is a linear continuous mapping between two Banach spaces and S and T are star -monotone operators. These lead to Brézis -Haraux -type approximation results for the range of the subdif-ferential of the sum between a proper convex lower -s...

We present a new duality theory to treat convex optimization problems and we prove that the geometric duality used by Scott and Jefferson in different papers during the last quarter of century is a special case of it. Moreover, weaker sufficient conditions to achieve strong duality are considered and optimality conditions are derived. Next, we appl...

We present some Farkas-type results for inequality systems involving finitely many convex constraints as well as convex max-functions. Therefore we use the dual of a minmax optimization problem. The main theorem and its consequences allows us to establish, as particular instances, some set containment characterizations and to rediscover two famous...

We present some Farkas-type results for inequality systems involving finitely many functions. Therefore we use a conjugate duality approach applied to an optimization problem with a composed convex objective function and convex inequality constraints. Some recently obtained results are rediscovered as special cases of our main result.

We used the so-called deterministic annealing algorithm due to Rose and Gurewitz by the classification of patent documents. A C++ program based on this algorithm was run first on some artificially constructed data and the results were good. Then we tested it on data sets obtained from some already classified patents. The conclusion we reached is th...

In this paper we present a new regularity condition for the subdifferential sum formula of a convex function with the precomposition of another convex function with a continuous linear mapping. This condition is formulated by using the epigraphs of the conjugates of the functions involved and turns out to be weaker than the generalized interior-poi...

We give an alternative formulation for the so-called closed cone constraint qualification (CCCQ) related to a convex optimization problem in Banach spaces recently introduced in the literature. This new formulation allows to prove in a simple way that (CCCQ) is weaker than some generalized interior-point constraint qualifications given in the past....

In this paper we provide a duality theory for multiobjective optimization problems with convex objective functions and finitely many D.C. constraints. In order to do this, we study first the duality for a scalar convex optimization problem with inequality constraints defined by extended real-valued convex functions. For a family of multiobjective p...

In this paper we deal with the construction of gap functions for equilibrium problems by using the Fenchel duality theory for convex optimization problems. For proving the properties which characterize a gap function weak and strong duality are used. Moreover, the proposed approach is applied to variational inequalities in a real Banach space.

It is not hard to prove that many convex optimization problems which are already studied in the literature can be rewritten as a particular instance of the following problem: minimize the sum of a convex function and the composition of a convex and K-increasing function with a K-convex one when the variable varies on a given set. Using a conjugate...

In this paper we present some duality assertions to a non-convex multiobjective fractional optimization problem. To the primal prob-lem we attach an intermediate multiobjective convex optimization problem, using an approach due to Dinkelbach ([6]), for which we construct then a dual problem. This is expressed in terms of the con-jugates of the nume...

We give a Brézis -Haraux -type approximation of the range of the monotone operator TA = A * • T • A when A is a linear continuous mapping between two Banach spaces and T is a maximal monotone operator. Then we specialize the result for a Brézis -Haraux -type approximation of the range of the subdifferential of the precomposition to A of a proper co...

We consider a convex optimization problem whose objective function consists of an entropy-like sum of functions k i=1 f i (x) ln(f i (x)/g i (x)). We calculate the Lagrange dual of this problem. Weak and strong duality assertions are presented, followed by the derivation of necessary and sufficient optimality conditions. Some entropy optimization p...

In this paper, strong duality for nearly-convex optimization problems is established. Three kinds of conjugate dual problems are associated to the primal optimization problem: the Lagrange dual, Fenchel dual, and Fenchel-Lagrange dual problems. The main result shows that, under suitable conditions, the optimal objective values of these four problem...

In the first part of our talk we give new Farkas-type results (theorems of the alternative) for inequality systems based on general conjugate duality assertions also developed within the presentation. Here we partially support ourselves on our recent paper (1). We derive also formulations of the Farkas-type results using the epigraphs of the occurr...

Since the huge database of patent documents is continuously increasing, the issue of classifying, updating and retrieving patent documents turned into an acute necessity. Therefore, we investigate the efficiency of applying Latent Semantic Indexing, an automatic indexing method of information retrieval, to some classes of patent documents from the...

In the first part of this study we introduced six different multiobjective dual problems to a general multiobjective optimization problem, for which we presented weak as well as strong duality assertions. Afterwards, we derived some inclusion results for the image sets of three of these problems. The aim of this second part is to complete our inves...

In this work we study the duality for a general multiobjective optimization problem. Considering, first, a scalar problem, different duals using the conjugacy approach are presented. Starting from these scalar duals, we introduce six different multiobjective dual problems to the primal one, one depending on certain vector parameters. The existence...

Having an optimization problem with linear objective function, linear inequality and maximum entropy inequality constraints, we determine a dual to it. Therefore we use a conjugacy approach which bases on the perturbation theory. As the main results we prove that the geometric dual problems introduced by Peterson for, both, unconstrained and constr...

In this paper we consider the optimization problem with a multiobjective com- posed convex function as objective function, namely, being a composite of a convex and componentwise increasing vector function with a convex vector function. By the conjugacy approach, we obtain a dual problem for it. The existence of weak and strong duality is proved.

We treat some duality assertions regarding multiobjective convex semidefinite pro-gramming problems. Having a vector minimization problem with convex entries in the objective vector function, we establish a dual for it using the so-called conjugacy approach. In order to deal with the duality assertions between these problems we need to study the du...

In this paper we consider, in a general normed space, the optimization problem with the objective function being a composite of a convex and componentwise increasing function with a vector convex function. Perturbing the primal problem, we obtain, by means of the Fenchel-Rockafellar approach, a dual problem for it. The existence of strong duality i...

This paper aims to extend duality investigations for the convex partially separable optimization problems. By using the results in [15] we formulate three dual problems for the optimization problem with convex inequality and affine equality constraints, which includes the convex partially separable one. For these duals we give a constraint qualific...

We present a text classification method based upon maximum entropy optimization. Having a set of documents which must be classified into some given classes, a maximum entropy optimization problem is considered. In order to solve this problem, we consider its Lagrange dual and we derive, by means of strong duality, the optimality conditions. After s...

In this paper we present a duality approach for a multiobjective fractional programming problem. The components of the vector objective function are particular ratios involving the square of a convex function and a positive concave function. Applying the Fenchel–Rockafellar duality theory for a scalar optimization problem associated to the multiobj...

Using the Fenchel-Rockafellar approach for the convex mathematical programming problem with inequality constraints different dual optimization problems by means of distinct perturbations of the primal problem are derived and studied. The classical Lagrange dual problem is one of those dual problems obtained by the perturbation of the right hand sid...

Using the Fenchel-Rockafellar approach for the convex mathematical programming problem with inequality constraints different dual optimization problems by means of distinct perturbations of the primal problem are derived and studied. The classical Lagrange dual problem is one of those dual problems obtained by the perturbation of the right hand sid...

We consider the classical Markowitz portfolio optimization problem with additional constraints representing so-called short sales.  The two objectives of this multiobjective problem are the expected return and the variance of a portfolio combined by a number of risky securities. A multiobjective problem is established which is dual to this classica...

A multiobjective programming problem characterized by convex goal functions and linear inequality constraints is studied. The investigation aims to the construction of a multiobjective dual problem permitting the verification of strong duality as well as optimality conditions. For the original primal problem properly efficient (minimal) solutions a...

A general convex multiobjective control approximation problem is considered with respect to duality. The single objectives contain linear functionals and powers of norms as parts, measuring the distance between linear mappings of the control variable and the state variables. Moreover, linear inequality constraints are included. A dual problem is es...

We consider a general vector minimum optimization problem with convex objective functions and linear inequality constraints. By means of linear scalarization and using the Fenchel-Rockafellar duality approach for the scalarized problem there is constructed a multiobjective dual maximum problem. The investigations are devoted to weak and strong dual...

Looking for m state variables and n control variables such that the sum of the distance functions between the state variables and the control variables becomes minimal is called control-approximation problem. This problem is investigated under constraints. Moreover, the distances between the control variables themselves are taken into account. Powe...

The classical Markowitz approach to portfolio selection leads to a biobjective optimization problem where the objectives are the expected return and the variance of a portfolio. In this paper a biobjective dual optimization problem to the Markowitz portfolio optimization problem is introduced and analyzed. For the Markowitz problem and its dual, we...

This paper provides a new method to obtain theorems of the alternative, using a location approach. This consists of associating a location problem with the original system of linear inequalities. The paper introduces a dual optimization problem and strong duality is established. From the dual problem, there is derived a second system of equations a...

Approximation problems with vecior–valued norms are considered, where The solution concept is that of proper efficiency in multicriterial optimization. From that one a notion of approximately efficient solutions is derived, so called ε–properly efficient solutions. Using Ekeland’s well known variational principle it results a slightly perturbed app...

The paper deals with multiobjective location problems of the so-called point-objective type with linear inequality restrictions and additional linear terms in the objective functions and linear operators in the norm parts. If the latter ones are replaced by the identity mapping one obtains the usual point-objective location problem. Approximately e...

In the present paper a general vectorial best approximation problem using vectorial norms with respect to properly efficient solutions is considered. Necessary and sufficient optimality conditions for such solutions are derived. This is done on base of scalarization and studying a corresponding dual problem to the scalar optimization problem. Also...

A general vectorial best approximation problem in linear and locally convex topological spaces, respectively, is considered. The approximation is based on socalled vectorial norms. For efficient, weakly efficient and strongly efficient solutions sufficient optimality conditions which can be interpreted as generalized Kolmogorov-conditions are obtai...

The paper is dealt with a general multicriterial location problems in a linear normed space. The objective vector function is composed of several objective functions us usual in scalar location. That is to minimize the sum of weighted distances between the new facility and a finite number of existing facilities. The individual goal functions differ...