Efficient solutions in V-KT-pseudoinvex multiobjective control problems: A characterization

In this paper, we introduce a new condition on functionals involved in a multiobjective control problem, for which we define the V-KT-pseudoinvex control problem. We prove that a V-KT-pseudoinvex control problem is characterized so that a Kuhn–Tucker point is an efficient solution. We generalize recently obtained optimality results of known mathematical programming problems and control problems. We illustrate these results with an example.     

Optimality and invexity in optimization problems in Banach algebras (spaces)

In this paper we introduce into nonsmooth optimization theory in Banach algebras a new class of mathematical programming problems, which generalizes the notion of smooth KT-(p,r)-invexity. In fact, this paper focuses on the optimality conditions for optimization problems in Banach algebras, regarding the generalized KT-(p,r)-invexity notion and Kuhn–Tucker points.     

Pseudoinvexity,optimality conditions and efficiency in multiobjective problems; duality

In this paper, we establish characterizations for efficient solutions to multiobjective programming problems, which generalize the characterization of established results for optimal solutions to scalar programming problems. So, we prove that in order for Kuhn–Tucker points to be efficient solutions it is necessary and sufficient that the multiobjective problem functions belong to a new class of functions, which we introduce. Similarly, we obtain characterizations for efficient solutions by using Fritz–John optimality conditions. Some examples are proposed to illustrate these classes of functions and optimality results. We study the dual problem and establish weak, strong and converse duality results.     

General constraint qualifications in nondifferentiable programming

We show that a familiar constraint qualification of differentiable programming has nonsmooth counterparts. As a result, necessary optimality conditions of Kuhn—Tucker type can be established for inequality-constrained mathematical programs involving functions not assumed to be differentiable, convex, or locally Lipschitzian. These optimality conditions reduce to the usual Karush—Kuhn—Tucker conditions in the differentiable case and sharpen previous results in the locally Lipschitzian case.     

Optimally Conditions for Pareto Nonsmooth Nonconvex Programming in Banach Spaces

In this paper, we consider a vector optimization problem where all functions involved are defined on Banach spaces. We obtain necessary and sufficient criteria for optimality in the form of Karush–Kuhn–Tucker conditions. We also introduce a nonsmooth dual problem and provide duality theorems.     

On optimality conditions in nondifferentiable programming

This paper is devoted to necessary optimality conditions in a mathematical programming problem without differentiability or convexity assumptions on the data. The main tool of this study is the concept of generalized gradient of a locally Lipschitz function (and more generally of a lower semi-continuous function). In the first part, we consider local extremization problems in the unconstrained case for objective functions taking values in (–, +]. In the second part, the constrained case is considered by the way of the cone of adherent displacements. In the presence of inequality constraints, we derive in the third part optimality conditions in the Kuhn—Tucker form under a constraint qualification.     

Generalized Kojima–Functions and Lipschitz Stability of Critical Points

In this paper we consider systems of equations which are defined by nonsmooth functions of a special structure. Functions of this type are adapted from Kojima's form of the Karush–Kuhn–Tucker conditions for C²—optimization problems. We shall show that such systems often represent conditions for critical points of variational problems (nonlinear programs, complementarity problems, generalized equations, equilibrium problems and others). Our main purpose is to point out how different concepts of generalized derivatives lead to characterizations of different Lipschitz properties of the critical point or the stationary solution set maps.     

A finite dimensional extension of Lyusternik theorem with applications to multiobjective optimization

We consider a multiobjective program with inequality and equality constraints and a set constraint. The equality constraints are Fréchet differentiable and the objective function and the inequality constraints are locally Lipschitz. Within this context, a Lyusternik type theorem is extended, establishing afterwards both Fritz–John and Kuhn–Tucker necessary conditions for Pareto optimality.     

Various definitions of the derivative in mathematical programming

This paper introduces seven derivatives in mathematical programming in locally convex topological vector spaces. All these derivatives have been known in various fields of mathematical sciences but they have never been used before in mathematical programming. The weakest of the seven derivatives is the compact derivative of Gil de Lamadrid and Sova. The derivative used by Neustadt in optimization theory is stronger than the compact derivative and it is equivalent to the derivative introduced by Michal and Bastiani. The main results of the paper show that the optimality conditions of both Lagrange—Kuhn—Tucker type and Caratheodory—John type hold for compactly differentiable functions. In the case of finite-dimensional spaces all these seven derivatives are equivalent to the Fréchet derivative.Research partly supported by the National Research Council of Canada. This paper was presented at the VIIIth International Symposium on Mathematical Programming held at Stanford University, August 27–31, 1973.     

On sufficiency and duality in multiobjective programming problem under generalized <Emphasis Type="Italic">α</Emphasis>-type I univexity

In this paper, we are concerned with the multiobjective programming problem with inequality constraints. We introduce new classes of generalized α-univex type I vector valued functions. A number of Kuhn–Tucker type sufficient optimality conditions are obtained for a feasible solution to be an efficient solution. The Mond–Weir type duality results are also presented.     

Expressions for subdifferential and optimization problems defined by nonsmooth homogeneous functions

In this paper, we prove a theoretical expression for subdifferentials of lower semicontinuous and homogeneous functions. The theoretical expression is a generalization of the Euler formula for differentiable homogeneous functions. As applications of the generalized Euler formula, we consider constrained optimization problems defined by nonsmooth positively homogeneous functions in smooth Banach spaces. Some results concerning Karush–Kuhn–Tucker points and necessary optimality conditions for the optimization problems are obtained.     

A Proof of the Necessity of Linear Independence Condition and Strong Second-Order Sufficient Optimality Condition for Lipschitzian Stability in Nonlinear Programming

For a nonlinear programming problem with a canonical perturbations, we give an elementary proof of the following result: If the Karush–Kuhn–Tucker map is locally single-valued and Lipschitz continuous, then the linear independence condition for the gradients of the active constraints and the strong second-order sufficient optimality condition hold.     

A penalty function method for solving inverse optimal value problem

In order to consider the inverse optimal value problem under more general conditions, we transform the inverse optimal value problem into a corresponding nonlinear bilevel programming problem equivalently. Using the Kuhn–Tucker optimality condition of the lower level problem, we transform the nonlinear bilevel programming into a normal nonlinear programming. The complementary and slackness condition of the lower level problem is appended to the upper level objective with a penalty. Then we give via an exact penalty method an existence theorem of solutions and propose an algorithm for the inverse optimal value problem, also analysis the convergence of the proposed algorithm. The numerical result shows that the algorithm can solve a wider class of inverse optimal value problem.     

Approximate Duality

We extend the Lagrangian duality theory for convex optimization problems to incorporate approximate solutions. In particular, we generalize well-known relationships between minimizers of a convex optimization problem, maximizers of its Lagrangian dual, saddle points of the Lagrangian, Kuhn–Tucker vectors, and Kuhn–Tucker conditions to incorporate approximate versions. As an application, we show how the theory can be used for convex quadratic programming and then apply the results to support vector machines from learning theory.     

A Modified SQP Method and Its Global Convergence

The sequential quadratic programming method developed by Wilson, Han andPowell may fail if the quadratic programming subproblems become infeasibleor if the associated sequence of search directions is unbounded. In [1], Hanand Burke give a modification to this method wherein the QP subproblem isaltered in a way which guarantees that the associated constraint region isnonempty and for which a robust convergence theory is established. In thispaper, we give a modification to the QP subproblem and provide a modifiedSQP method. Under some conditions, we prove that the algorithm eitherterminates at a Kuhn–Tucker point within finite steps or generates aninfinite sequence whose every cluster is a Kuhn–Tucker point.Finally, we give some numerical examples.     

Generalized Convexity and Nonsmooth Problems of Vector Optimization

In this paper it is shown that every generalized Kuhn-Tucker point of a vector optimization problem involving locally Lipschitz functions is a weakly efficient point if and only if this problem is KT- pseudoinvex in a suitable sense. Under a closedness assumption (in particular, under a regularity condition of the constraint functions) it is pointed out that in this result the notion of generalized Kuhn–Tucker point can be replaced by the usual notion of Kuhn–Tucker point. Some earlier results in (Martin (1985), The essence of invexity, J. Optim. Theory Appl., 47, 65–76. Osuna-Gómez et al., (1999), J. Math. Anal. Appl., 233, 205–220. Osuna-GGómez et al., (1998), J. Optim. Theory Appl., 98, 651–661. Phuong et al., (1995) J. Optim. Theory Appl., 87, 579–594) results are included as special cases of ours. The paper also contains characterizations of HC-invexity and KT- invexity properties which are sufficient conditions for KT- pseudoinvexity property of nonsmooth problems.Mathematics Subject Classifications: 90C29, 26B25     

Applications of symmetric derivatives in mathematical programming

In recent times the Kuhn—Tucker optimality conditions and the duality theorems for convex programming have been extended by generalizations of the convexity concept. In this paper the notion of a symmetric derivative for a function of several variables is introduced and used to provide extensions of some fundamental optimality and duality theorems of convex programming. Symmetric derivatives are also used to extend some optimality and duality theorems involving pseudoconvexity and differentiable quasiconvexity.     

On the Complexity of Equalizing Inequalities

We study two approaches to replace a finite mathematical programming problem with inequality constraints by a problem that contains only equality constraints. The first approach lifts the feasible set into a high-dimensional space by the introduction of quadratic slack variables. We show that then not only the number of critical points but also the topological complexity of the feasible set grow exponentially. On the other hand, the second approach bases on an interior point technique and lifts an approximation of the feasible set into a space with only one additional dimension. Here only Karush–Kuhn–Tucker points with respect to the positive and negative objective function in the original problem give rise to critical points of the smoothed problem, so that the number of critical points as well as the topological complexity can at most double.     

A Simply Constrained Optimization Reformulation of KKT Systems Arising from Variational Inequalities

The Karush—Kuhn—Tucker (KKT) conditions can be regarded as optimality conditions for both variational inequalities and constrained optimization problems. In order to overcome some drawbacks of recently proposed reformulations of KKT systems, we propose casting KKT systems as a minimization problem with nonnegativity constraints on some of the variables. We prove that, under fairly mild assumptions, every stationary point of this constrained minimization problem is a solution of the KKT conditions. Based on this reformulation, a new algorithm for the solution of the KKT conditions is suggested and shown to have some strong global and local convergence properties. Accepted 10 December 1997     

Implicit function theorems for mathematical programming and for systems of inequalities

Implicit function formulas for differentiating the solutions of mathematical programming problems satisfying the conditions of the Kuhn—Tucker theorem are motivated and rigorously demonstrated. The special case of a convex objective function with linear constraints is also treated with emphasis on computational details. An example, an application to chemical equililibrium problems, is given.Implicit function formulas for differentiating the unique solution of a system of simultaneous inequalities are also derived.