On necessary optimality conditions in vector optimization problems

Necessary conditions of the multiplier rule type for vector optimization problems in Banach spaces are proved by using separation theorems and Ljusternik's theorem. The Pontryagin maximum principle for multiobjective control problems with state constraints is derived from these general conditions. The paper extends to vector optimization results established in the scalar case by Ioffe and Tihomirov.     

On optimality conditions for quasi-relative efficient solutions in set-valued optimization

In the paper, we establish necessary and sufficient optimality conditions for quasi-relative efficient solutions of a constrained set-valued optimization problem using the Lagrange multipliers. Many examples are given to show that our results and their applications are more advantageous than some existing ones in the literature.     

On necessary and sufficient optimality conditions for multicriterial optimization problems

In this paper we derive first order necessary and sufficient optimality conditions for nonsmooth optimization problems with multiple criteria. These conditions are given for different optimality notions (i.e. weak, Pareto- and proper minimality) and for different types of derivatives of nonsmooth objective functions (locally Lipschitz continuous and quasidifferentiable) mappings. The conditions are given, if possible, in terms of a derivative and a subdifferential of those mappings.     

On optimality conditions for multiobjective optimization problems in topological vector space

In this paper, we are concerned with a differentiable multiobjective programming problem in topological vector spaces. An alternative theorem for generalized K subconvexlike mappings is given. This permits the establishment of optimality conditions in this context: several generalized Fritz John conditions, in line to those in Hu and Ling [Y. Hu, C. Ling, The generalized optimality conditions of multiobjective programming problem in topological vector space, J. Math. Anal. Appl. 290 (2004) 363-372] are obtained and, in the presence of the generalized Slater's constraint qualification, the Karush-Kuhn-Tucker necessary optimality conditions.     

On reductibility of degenerate optimization problems to regular operator equations

We present an application of the p-regularity theory to the analysis of non-regular (irregular, degenerate) nonlinear optimization problems. The p-regularity theory, also known as the p-factor analysis of nonlinear mappings, was developed during last thirty years. The p-factor analysis is based on the construction of the p-factor operator which allows us to analyze optimization problems in the degenerate case. We investigate reducibility of a non-regular optimization problem to a regular system of equations which do not depend on the objective function. As an illustration we consider applications of our results to non-regular complementarity problems of mathematical programming and to linear programming problems.     

Necessary optimality conditions for bilevel set optimization problems

Bilevel programming problems are hierarchical optimization problems where in the upper level problem a function is minimized subject to the graph of the solution set mapping of the lower level problem. In this paper necessary optimality conditions for such problems are derived using the notion of a convexificator by Luc and Jeyakumar. Convexificators are subsets of many other generalized derivatives. Hence, our optimality conditions are stronger than those using e.g., the generalized derivative due to Clarke or Michel-Penot. Using a certain regularity condition Karush-Kuhn-Tucker conditions are obtained.      

Sequential optimality conditions for composed convex optimization problems

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 function with a K-convex and K-epi-closed (continuous) function, where K is a nonempty convex cone. We prove that several results from the literature dealing with sequential characterizations of subgradients are obtained as particular cases of our results. We also improve the above mentioned statements.     

Fuzzy necessary optimality conditions for vector optimization problems

The primary aim of this article is to derive Lagrange multiplier rules for vector optimization problems using a non-convex separation technique and the concept of abstract subdifferential. Furthermore, we present a method of estimation of the norms of such multipliers in very general cases and for many particular subdifferentials.     

On second-order optimality conditions in constrained multiobjective optimization

In this paper we study a multiobjective optimization problem with inequality constraints on finite dimensional spaces. A second-order necessary condition for local weak efficiency is proved under strict differentiability assumptions. We also establish a second-order sufficient condition for local firm efficiency of order 2 under ?-stability assumptions. In this way we generalize some corresponding results obtained by P.Q. Khanh and N.D. Tuan, and by the authors.     

Sequential optimality conditions for cardinality-constrained optimization problems with applications

Computational Optimization and Applications - Recently, a new approach to tackle cardinality-constrained optimization problems based on a continuous reformulation of the problem was proposed....     

Strict directional solutions in vectorial problems: necessary optimality conditions

Journal of Global Optimization - We study directional strict efficiency in vector optimization and equilibrium problems with set-valued map objectives. We devise several possibilities to define a...     

Efficient solutions and optimality conditions for vector equilibrium problems

Necessary optimality conditions for efficient solutions of unconstrained and vector equilibrium problems with equality and inequality constraints are derived. Under assumptions on generalized convexity, necessary optimality conditions for efficient solutions become sufficient optimality conditions. Note that it is not required here that the ordering cone in the objective space has a nonempty interior.     

Second-order optimality conditions for cone-subarcwise connected set-valued optimization problems

The concept of a cone subarcwise connected set-valued map is introduced. Several examples are given to illustrate that the cone subarcwise connected set-valued map is a proper generalization of the cone arcwise connected set-valued map, as well as the arcwise connected set is a proper generalization of the convex set, respectively. Then, by virtue of the generalized second-order contingent epiderivative, second-order necessary optimality conditions are established for a point pair to be a local global proper efficient element of set-valued optimization problems. When objective function is cone subarcwise connected, a second-order sufficient optimality condition is also obtained for a point pair to be a global proper efficient element of set-valued optimization problems.     

Second order optimality conditions for bilevel set optimization problems

In this paper, we introduce a new notion of augmenting function known as indicator augmenting function to establish a minmax type duality relation, existence of a path of solution converging to optimal value and a zero duality gap relation for a nonconvex primal problem and the corresponding Lagrangian dual problem. We also obtain necessary and sufficient conditions for an exact penalty representation in the framework of indicator augmented Lagrangian.     

On analytic solutions of operator differential equations

We find conditions on a closed operator A in a Banach space that are necessary and sufficient for the existence of solutions of a differential equation y′(t) = Ay(t), t ∈[0,∞),in the classes of entire vector functions with given order of growth and type. We present criteria for the denseness of classes of this sort in the set of all solutions. These criteria enable one to prove the existence of a solution of the Cauchy problem for the equation under consideration in the class of analytic vector functions and to justify the convergence of the approximate method of power series. In the special case where A is a differential operator, the problem of applicability of this method was first formulated by Weierstrass. Conditions under which this method is applicable were found by Kovalevskaya.     

On holomorphic solutions of some boundary-value problems for second-order elliptic operator differential equations

In the class of holomorphic vector functions, we determine the conditions of solvability of the boundaryvalue problem for a class of second-order operator differential equations expressed in terms of the operator coefficients appearing both in the equation and in the boundary condition.     

On optimality conditions of relaxed non-convex variational problems

Non-convex variational problems in many situations lack a classical solution. Still they can be solved in a generalized sense, e.g., they can be relaxed by means of Young measures. Various sets of optimality conditions of the relaxed non-convex variational problems can be introduced. For example, the so-called “variations” of Young measures lead to a set of optimality conditions, or the Weierstrass maximum principle can be the base of another set of optimality conditions. Moreover the second order necessary and sufficient optimality conditions can be derived from the geometry of the relaxed problem. In this article the sets of optimality conditions are compared. Illustrative examples are included.