Linear Vector Optimization Research Articles

Stability and Scalarization in Vector Optimization Using Improvement Sets

The aim of this paper is to study certain aspects of stability and scalarization of a nonconvex vector optimization problem through improvement sets. This paper attempts to investigate an open problem on stability posed by Chicco et al. The notion of stability is studied through Painleve---Kuratowski set-convergence, where we establish sufficiency conditions for the lower and upper set-convergences of optimal solution sets of a family of perturbed vector problems, both in the given space and its image space. The perturbations are performed both on the objective function and the feasible set. Further, by using a nonlinear scalarization function defined in terms of an improvement set, we establish lower and upper Painleve---Kuratowski set-convergences of sequences of approximate solution sets of certain scalarized problems. We then link these set-convergences with the set-convergences of optimal solution sets of the perturbed problems. Finally, we investigate the stability and scalarization of a linear vector optimization problem in finite dimensional spaces.

AN ALGORITHM FOR CALCULATING THE SET OF SUPERHEDGING PORTFOLIOS IN MARKETS WITH TRANSACTION COSTS

We study the explicit calculation of the set of superhedging portfolios of contingent claims in a discrete-time market model for d assets with proportional transaction costs. The set of superhedging portfolios can be obtained by a recursive construction involving set operations, going backward in the event tree. We reformulate the problem as a sequence of linear vector optimization problems and solve it by adapting known algorithms. The corresponding superhedging strategy can be obtained going forward in the tree. Examples are given involving multiple correlated assets and basket options. Furthermore, we relate existing algorithms for the calculation of the scalar superhedging price to the set-valued algorithm by a recent duality theory for vector optimization problems. The main contribution of the paper is to establish the connection to linear vector optimization, which allows to solve numerically multi-asset superhedging problems under transaction costs.

Benson type algorithms for linear vector optimization and applications

New versions and extensions of Benson's outer approximation algorithm for solving linear vector optimization problems are presented. Primal and dual variants are provided in which only one scalar linear program has to be solved in each iteration rather than two or three as in previous versions. Extensions are given to problems with arbitrary pointed solid polyhedral ordering cones. Numerical examples are provided, one of them involving a new set-valued risk measure for multivariate positions.

The Pascoletti–Serafini Scalarization Scheme and Linear Vector Optimization

The Pascoletti---Serafini scalarization scheme for general vector optimization problems is studied. It is specified to linear vector optimization to give minimal representation formulae for the weakly efficient solution set and the efficient solution set. Several facts on connectedness of the solution sets of Pascoletti---Serafini's scalar auxiliary problems, both for linear vector optimization and for nonlinear vector optimization, are established.

An algorithm to solve polyhedral convex set optimization problems

An algorithm which computes a solution of a set optimization problem is provided. The graph of the objective map is assumed to be given by finitely many linear inequalities. A solution is understood to be a set of points in the domain satisfying two conditions: the attainment of the infimum and minimality with respect to a set relation. In the first phase of the algorithm, a linear vector optimization problem, called the vectorial relaxation, is solved. The resulting pre-solution yields the attainment of the infimum but, in general, not minimality. In the second phase of the algorithm, minimality is established by solving certain linear programs in combination with vertex enumeration of some values of the objective map.

A property of bicriteria affine vector variational inequalities

By a scalarization method, it is proved that both the Pareto solution set and the weak Pareto solution set of a bicriteria affine vector variational inequality have finitely many connected components provided that a regularity condition is satisfied. An explicit upper bound for the numbers of connected components of the Pareto solution set and the weak Pareto solution set is obtained. Consequences of the results for bicriteria quadratic vector optimization problems and linear fractional vector optimization problems are discussed in detail. Under an additional assumption on the data set, Theorems 3.1 and 3.2 in this article solve in the affirmative Question 1 in Yen and Yao [N.D. Yen and J.-C. Yao, Monotone affine vector variational inequalities, Optimization 60 (2011), pp. 53–68] and Question 9.3 in Yen [N.D. Yen, Linear fractional and convex quadratic vector optimization problems, in Recent Developments in Vector Optimization, Q.H. Ansari and J.-C. Yao, eds, Springer-Verlag, New York, 2012, pp. 297–328] for the case of bicriteria problems without requiring the monotonicity. Besides, the theorems also give a partial solution to Question 2 in Yen and Yao (2011) about finding an upper bound for the numbers of connected components of the solution sets under investigation.

Monotone affine vector variational inequalities

By using a stability theorem of S.M. Robinson and a scalarization method, we establish sufficient conditions for the upper semicontinuity of the solution maps of parametric monotone affine vector variational inequalities. As a by-product, some new topological properties of the solution sets of these problems are obtained. Our results imply several facts for the solution stability and connectedness of the solution sets of convex quadratic vector optimization problems and of linear fractional vector optimization problems.

On linear vector optimization duality in infinite-dimensional spaces

In this paper we extend to infinite-dimensional spaces a vector duality concept recently considered in the literature in connection to the classical vector minimization linear optimization problem in a finite-dimensional framework. Weak, strong and converse duality for the vector dual problem introduced with this respect are proven and we also investigate its connections to some classical vector duals considered in the same framework in the literature.

Classical linear vector optimization duality revisited

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 vector objective function. Weak, strong and converse duality for this "new-old" vector dual problem are proven and we also investigate its connections to other vector duals considered in the same framework in the literature. We also show that the efficient solutions of the classical linear vector optimization problem coincide with its properly efficient solutions (in any sense) when the image space is partially ordered by a nontrivial pointed closed convex cone, too.

On linear vector program and vector matrix game equivalence

We define a vector matrix game, which is a vector version of the usual matrix game and consists of more than two skew symmetric matrices and vector ordering. Using vector optimization techniques, we characterize solutions for the vector matrix game. We establish equivalent relations between a linear vector optimization problem and its corresponding vector matrix game.

Pseudo-Lipschitz property of linear semi-infinite vector optimization problems

This paper is devoted to the study of the pseudo-Lipschitz property of Pareto solution map for the parametric linear semi-infinite vector optimization problem (LSVO). We establish new sufficient conditions for the pseudo-Lipschitz property of the Pareto solution map of (LSVO) under continuous perturbations of the right-hand side of the constraints and linear perturbations of the objective function. Examples are given to illustrate the results obtained.

Pareto Solutions of Polyhedral-valued Vector Optimization Problems in Banach Spaces

In general Banach spaces, we consider a vector optimization problem (SVOP) in which the objective is a set-valued mapping whose graph is the union of finitely many polyhedra. We establish some results on structure and connectedness of the weak Pareto solution set, Pareto solution set, weak Pareto optimal value set and Pareto optimal value set of (SVOP). In particular, we improve and generalize Arrow, Barankin and Blackwell’s classical results on linear vector optimization problems in Euclidean spaces.

An exact penalty on bilevel programs with linear vector optimization lower level

We are interested in a class of linear bilevel programs where the upper level is a linear scalar optimization problem and the lower level is a linear multi-objective optimization problem. We approach this problem via an exact penalty method. Then, we propose an algorithm illustrated by numerical examples.

ON OPTIMALITY CONDITIONS FOR ABSTRACT CONVEX VECTOR OPTIMIZATION PROBLEMS

A sequential optimality condition characterizing the efficient solution without any constraint qualification for an abstract convex vector optimization problem is given in sequential forms using subdifferentials and <TEX>${\epsilon}$</TEX>-subdifferentials. Another sequential condition involving only the subdifferentials, but at nearby points to the efficient solution for constraints, is also derived. Moreover, we present a proposition with a sufficient condition for an efficient solution to be properly efficient, which are a generalization of the well-known Isermann result for a linear vector optimization problem. An example is given to illustrate the significance of our main results. Also, we give an example showing that the proper efficiency may not imply certain closeness assumption.

A new approach to duality in vector optimization

In this article, we develop a new approach to duality theory for convex vector optimization problems. We modify a given (set-valued) vector optimization problem such that the image space becomes a complete lattice (a sublattice of the power set of the original image space), where the corresponding infimum and supremum are sets that are related to the set of (minimal and maximal) weakly efficient points. In doing so we can carry over the structures of the duality theory in scalar convex programming. Exemplarily this is demonstrated for the case of Fenchel duality. We also show the relationship to set-valued optimization based on the ordering ‘set inclusion’. Finally, some consequences for duality in linear vector optimization are discussed.

Unbounded Components in the Solution Sets of Strictly Quasiconcave Vector Maximization Problems

Let (P) denote the vector maximization problem $$\max\{f(x)=\big(f_1(x),\ldots,f_m(x)\big){:}\,x\in D\},$$ where the objective functions f i are strictly quasiconcave and continuous on the feasible domain D, which is a closed and convex subset of R n . We prove that if the efficient solution set E(P) of (P) is closed, disconnected, and it has finitely many (connected) components, then all the components are unbounded. A similar fact is also valid for the weakly efficient solution set E w (P) of (P). Especially, if f i (i=1,...,m) are linear fractional functions and D is a polyhedral convex set, then each component of E w (P) must be unbounded whenever E w (P) is disconnected. From the results and a result of Choo and Atkins [J. Optim. Theory Appl. 36, 203---220 (1982.)] it follows that the number of components in the efficient solution set of a bicriteria linear fractional vector optimization problem cannot exceed the number of unbounded pseudo-faces of D.

Linear fractional vector optimization problems with many components in the solution sets

Linear fractional vector optimization (LFVO) problems form a special class of nonconvex multiobjective optimization problems which has a significant role both in the management science and in the theory of vector optimization. Up to now, only LFVO problems with at most two connected components in the solution sets have been discussed in the literature. We propose some examples of LFVO problems with three or more connected components in the solution sets. It is proved that for any integer $m$ there exist LFVO problems with $m$ objective criteria whose solution sets have exactly $m$ connected components. Besides, we have solved the conjecture saying that $\chi(E(\mbox{P}))\leq \min\{m,\mbox{dim}0^+D+1\},$ where $\chi(E(\mbox{P}))$ is the number of connected components in the efficient solution set of a LFVO problem $(\mbox{P})$, $m$ is the number of the objective criteria of $(\mbox{P})$, and $\mbox{dim}0^+D$ is the dimension of the recession cone $0^+D$ of the feasible domain $D$ of $(\mbox{P})$. These new facts are useful for analyzing the practical problems which can be modeled as quasiconcave vector maximization problems in general, and as LFVO problems on unbounded feasible domains in particular.

Stability of linear vector optimization problems corresponding to an efficient set

This paper deals with the set of all parameters corresponding to the set of all efficient points of parametric linear multiobjective programming problems. Here, we consider two classes of parametric optimization problems one of them is parameters in the objective functions and the other is parameters in the right-hand side of the constraints. An algorithm for determining this set is given. Some illustrative examples are given to clarify this algorithm.

A Parametrical Generalized Solution in Linear Vector Optimization

A parametrical representation of a linear problem of Pareto optimization is considered. The notion of a generalized solution is introduced for an appropriate pair of dual problems. The structure of a parametrical generalized solution is investigated. A relation between generalized and efficient solutions of problems is established.

Multiobjective Control Approximation Problems: Duality and Optimality

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 established, and weak and strong duality properties as well as necessary and sufficient optimality conditions are derived. Point-objective location problems and linear vector optimization problems turn out to be special cases of the problem investigated. Therefore, well-known duality results for linear vector optimization are obtained as special cases.