Multiobjective optimization problems with equilibrium constraints

The paper is devoted to new applications of advanced tools of modern variational analysis and generalized differentiation to the study of broad classes of multiobjective optimization problems subject to equilibrium constraints in both finite-dimensional and infinite-dimensional settings. Performance criteria in multiobjective/vector optimization are defined by general preference relationships satisfying natural requirements, while equilibrium constraints are described by parameterized generalized equations/variational conditions in the sense of Robinson. Such problems are intrinsically nonsmooth and are handled in this paper via appropriate normal/coderivative/subdifferential constructions that exhibit full calculi. Most of the results obtained are new even in finite dimensions, while the case of infinite-dimensional spaces is significantly more involved requiring in addition certain “sequential normal compactness” properties of sets and mappings that are preserved under a broad spectrum of operations.     

Multiobjective symmetric duality with cone constraints

We formulate a pair of multiobjective symmetric dual programs for pseudo-invex functions and arbitrary cones. Our model is unifying the Wolfe vector symmetric dual and the Mond-Weir vector symmetric dual models. We establish the weak, strong, converse and self duality theorems for our pair of dual models. Nanda and Das' results (Optimization 28 (1994) 267; Eur. J. Oper. Res. 88 (1996) 572) are obtained as special cases.     

The optimality conditions for optimization problems with convex constraints and multiple fuzzy-valued objective functions

The optimality conditions for multiobjective programming problems with fuzzy-valued objective functions are derived in this paper. The solution concepts for these kinds of problems will follow the concept of nondominated solution adopted in the multiobjective programming problems. In order to consider the differentiation of fuzzy-valued functions, we invoke the Hausdorff metric to define the distance between two fuzzy numbers and the Hukuhara difference to define the difference of two fuzzy numbers. Under these settings, the optimality conditions for obtaining the (strongly, weakly) Pareto optimal solutions are elicited naturally by introducing the Lagrange multipliers.     

Duality for multiobjective optimization problems with convex objective functions and D.C. constraints

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 problems associated to the initial one we determine then, by means of the scalar duality results, their multiobjective dual problems. Finally, we consider as a special case the duality for the convex multiobjective optimization problem with convex constraints.     

The optimality conditions for optimization problems with fuzzy-valued objective functions

The optimality conditions for an optimization problem with fuzzy-valued objective function are derived in this article. The solution concept of this optimization problem will follow the similar solution concept, called nondominated solution, in multiobjective programming problem. In order to consider the differentiation of fuzzy-valued function, we invoke the Hausdorff metric to define the distance between two fuzzy numbers and the Hukuhara difference to define the difference of two fuzzy numbers. Under these settings, the optimality conditions for obtaining the nondominated solutions are elicited naturally by introducing the Lagrange multipliers.     

Multiobjective optimization problem with variational inequality constraints

 We study a general multiobjective optimization problem with variational inequality, equality, inequality and abstract constraints. Fritz John type necessary optimality conditions involving Mordukhovich coderivatives are derived. They lead to Kuhn-Tucker type necessary optimality conditions under additional constraint qualifications including the calmness condition, the error bound constraint qualification, the no nonzero abnormal multiplier constraint qualification, the generalized Mangasarian-Fromovitz constraint qualification, the strong regularity constraint qualification and the linear constraint qualification. We then apply these results to the multiobjective optimization problem with complementarity constraints and the multiobjective bilevel programming problem. Received: November 2000 / Accepted: October 2001 Published online: December 19, 2002 Key Words. Multiobjective optimization – Variational inequality – Complementarity constraint – Constraint qualification – Bilevel programming problem – Preference – Utility function – Subdifferential calculus – Variational principle Research of this paper was supported by NSERC and a University of Victoria Internal Research Grant Research was supported by the National Science Foundation under grants DMS-9704203 and DMS-0102496 Mathematics Subject Classification (2000): Sub49K24, 90C29     

Equivalence and existence of weak Pareto optima for multiobjective optimization problems with cone constraints

In this work, a differentiable multiobjective optimization problem with generalized cone constraints is considered, and the equivalence of weak Pareto solutions for the problem and for its η-approximated problem is established under suitable conditions. Two existence theorems for weak Pareto solutions for this kind of multiobjective optimization problem are proved by using a Karush–Kuhn–Tucker type optimality condition and the F-KKM theorem.     

Strongly sub-feasible direction method for constrained optimization problems with nonsmooth objective functions

In this paper, we propose a strongly sub-feasible direction method for the solution of inequality constrained optimization problems whose objective functions are not necessarily differentiable. The algorithm combines the subgradient aggregation technique with the ideas of generalized cutting plane method and of strongly sub-feasible direction method, and as results a new search direction finding subproblem and a new line search strategy are presented. The algorithm can not only accept infeasible starting points but also preserve the “strong sub-feasibility” of the current iteration without unduly increasing the objective value. Moreover, once a feasible iterate occurs, it becomes automatically a feasible descent algorithm. Global convergence is proved, and some preliminary numerical results show that the proposed algorithm is efficient.     

Quasi-relative interior-type constraint qualifications ensuring strong Lagrange duality for optimization problems with cone and affine constraints

In this article we provide weak sufficient strong duality conditions for a convex optimization problem with cone and affine constraints, stated in infinite dimensional spaces, and its Lagrange dual problem. Our results are given by using the notions of quasi-relative interior and quasi-interior for convex sets. The main strong duality theorem is accompanied by several stronger, yet easier to verify in practice, versions of it. As exemplification we treat a problem which is inspired from network equilibrium. Our results come as corrections and improvements to Daniele and Giuffré (2007) [9].     

Scalarization of constraints system in some vector optimization problems and applications

In this work we employ a new method to penalize a constrained non solid vector optimization problem by means of a scalarization functional applied to the constraints system. Then, we formulate optimality conditions which mainly use several types of regularity for single and set-valued maps. In order to motivate our demarche, we discuss in detail the assumptions used in the main results and we show how it can be verified.     

A random coordinate descent algorithm for optimization problems with composite objective function and linear coupled constraints

In this paper we propose a variant of the random coordinate descent method for solving linearly constrained convex optimization problems with composite objective functions. If the smooth part of the objective function has Lipschitz continuous gradient, then we prove that our method obtains an ?-optimal solution in $\mathcal{O}(n^{2}/\epsilon)$ iterations, where n is the number of blocks. For the class of problems with cheap coordinate derivatives we show that the new method is faster than methods based on full-gradient information. Analysis for the rate of convergence in probability is also provided. For strongly convex functions our method converges linearly. Extensive numerical tests confirm that on very large problems, our method is much more numerically efficient than methods based on full gradient information.     

Mathematical programming problems with quasi-convex objective functions

This paper generalizes results obtained by Arrow and Enthoven in [1] for functions of ClassC ² on the non-negative orthant.This research was supported by Hydro—Quebec; University of Montreal; Office of Naval Research, Contract N-0014-67-A0112-001; National Science Foundation, Grant GP 25738.     

New properties and computational improvement of the GOP algorithm for problems with quadratic objective functions and constraints

In Floudas and Visweswaran (1990, 1993), a deterministic global optimization approach was proposed for solving certain classes of nonconvex optimization problems. An algorithm, GOP, was presented for the solution of the problem through a series ofprimal andrelaxed dual problems that provide valid upper and lower bounds respectively on the global solution. The algorithm was proved to have finite convergence to an -global optimum. In this paper, new theoretical properties are presented that help to enhance the computational performance of the GOP algorithm applied to problems of special structure. The effect of the new properties is illustrated through application of the GOP algorithm to a difficult indefinite quadratic problem, a multiperiod tankage quality problem that occurs frequently in the modeling of refinery processes, and a set of pooling/blending problems from the literature. In addition, extensive computational experience is reported for randomly generated concave and indefinite quadratic programming problems of different sizes. The results show that the properties help to make the algorithm computationally efficient for fairly large problems.     

Vector optimization problems with quasiconvex constraints

Let X be a real linear space, a convex set, Y and Z topological real linear spaces. The constrained optimization problem min _C f(x), is considered, where f : X ₀→ Y and g : X ₀→ Z are given (nonsmooth) functions, and and are closed convex cones. The weakly efficient solutions (w-minimizers) of this problem are investigated. When g obeys quasiconvex properties, first-order necessary and first-order sufficient optimality conditions in terms of Dini directional derivatives are obtained. In the special case of problems with pseudoconvex data it is shown that these conditions characterize the global w-minimizers and generalize known results from convex vector programming. The obtained results are applied to the special case of problems with finite dimensional image spaces and ordering cones the positive orthants, in particular to scalar problems with quasiconvex constraints. It is shown, that the quasiconvexity of the constraints allows to formulate the optimality conditions using the more simple single valued Dini derivatives instead of the set valued ones.      

On optimization problems with set-valued objective maps

In this paper, we consider constrained optimization problems with set-valued objective maps. First, we define three types of quasi orderings on the set of all non-empty subsets in n-dimensional Euclidean space and investigate their properties. Next, by using these orderings, we define the concepts of the convexities to set-valued maps and investigate their properties. Finally, based on these results, we define the concepts of optimal solutions to constrained optimization problems with set-valued objective maps and characterize their properties.     

A coverage-based Box-Algorithm to compute a representation for optimization problems with three objective functions

In this paper, a method for computing a substitute of the nondominated set, a so-called representation system, is proposed for tricriteria optimization problems. Our algorithm is easy to implement, it is flexibly applicable to a wide range of problems, and it takes several quality measures which have been proposed in the literature into account. The resulting representation system satisfies provably certain quality levels and, thus, it may be utilized to make proper and confident decisions.     

一类锥约束多目标优化问题的高阶对偶研究

在一类锥约束单目标优化问题的一阶对偶模型基础之上,建立了锥约束多目标优化问题的二阶和高阶对偶模型.在广义凸性假设下,给出了弱对偶定理,在Kuhn-Tucker约束品性下,得到了强对偶定理.最后,在弱对偶定理的基础上,利用Fritz-John型必要条件建立了逆对偶定理.     

Symmetric duality for fractional variational problems with cone constraints

A pair of symmetric fractional variational programming problems is formulated over cones. Weak, strong, converse and self duality theorems are discussed under pseudoinvexity. Static symmetric dual fractional programs are included as special case and corresponding symmetric duality results are merely stated.     

On multiple objective programming problems with set functions

The convexity of a subset of a σ-algebra and the convexity of a set function on a convex subset are defined. Related properties are also examined. A Farkas-Minkowski theorem for set functions is then proved. These results are used to characterize properly efficient solutions for multiple objective programming problems with set functions by associated scalar problems.     

Solving combinatorial problems with combined Min-Max-Min-Sum objective and applications

The purpose of this paper is to introduce and study a new class of combinatorial optimization problems in which the objective function is the algebraic sum of a bottleneck cost function (Min-Max) and a linear cost function (Min-Sum). General algorithms for solving such problems are described and general complexity results are derived. A number of examples of application involving matchings, paths and cutsets, matroid bases, and matroid intersection problems are examined, and the general complexity results are specialized to each of them. The interest of these various problems comes in particular from their strong relation to other important and difficult combinatorial problems such as: weighted edge coloring of a graph; optimum weighted covering with matroid bases; optimum weighted partitioning with matroid intersections, etc. Another important area of application of the algorithms given in the paper is bicriterion analysis involving a Min-Max criterion and a Min-Sum one.