Second-order Karush–Kuhn–Tucker optimality conditions for set-valued optimization

In this paper, we propose the concept of a second-order composed contingent derivative for set-valued maps, discuss its relationship to the second-order contingent derivative and investigate some of its special properties. By virtue of the second-order composed contingent derivative, we extend the well-known Lagrange multiplier rule and the Kurcyusz–Robinson–Zowe regularity assumption to a constrained set-valued optimization problem in the second-order case. Simultaneously, we also establish some second-order Karush–Kuhn–Tucker necessary and sufficient optimality conditions for a set-valued optimization problem, whose feasible set is determined by a set-valued map, under a generalized second-order Kurcyusz–Robinson–Zowe regularity assumption.     

The Karush–Kuhn–Tucker optimality conditions for fuzzy optimization problems

This paper considers optimization problems with fuzzy-valued objective functions. For this class of fuzzy optimization problems we obtain Karush–Kuhn–Tucker type optimality conditions considering the concept of generalized Hukuhara differentiable and pseudo-invex fuzzy-valued functions.     

The Karush–Kuhn–Tucker optimality conditions in multiobjective programming problems with interval-valued objective functions

The KKT conditions in multiobjective programming problems with interval-valued objective functions are derived in this paper. Many concepts of Pareto optimal solutions are proposed by considering two orderings on the class of all closed intervals. In order to consider the differentiation of an interval-valued function, we invoke the Hausdorff metric to define the distance between two closed intervals and the Hukuhara difference to define the difference of two closed intervals. Under these settings, we are able to consider the continuity and differentiability of an interval-valued function. The KKT optimality conditions can then be naturally elicited.     

Enhanced Karush–Kuhn–Tucker Conditions for Mathematical Programs with Equilibrium Constraints

In this paper, we study necessary optimality conditions for nonsmooth mathematical programs with equilibrium constraints. We first show that, unlike the smooth case, the mathematical program with equilibrium constraints linear independent constraint qualification is not a constraint qualification for the strong stationary condition when the objective function is nonsmooth. We then focus on the study of the enhanced version of the Mordukhovich stationary condition, which is a weaker optimality condition than the strong stationary condition. We introduce the quasi-normality and several other new constraint qualifications and show that the enhanced Mordukhovich stationary condition holds under them. Finally, we prove that quasi-normality with regularity implies the existence of a local error bound.     

Geometric conditions for Kuhn–Tucker sufficiency of global optimality in mathematical programming

We present geometric criteria for a feasible point that satisfies the Kuhn–Tucker conditions to be a global minimizer of mathematical programming problems with or without bounds on the variables. The criteria apply to multi-extremal programming problems which may have several local minimizers that are not global. We establish such criteria in terms of underestimators of the Lagrangian of the problem. The underestimators are required to satisfy certain geometric property such as the convexity (or a generalized convexity) property. We show that the biconjugate of the Lagrangian can be chosen as a convex underestimator whenever the biconjugate coincides with the Lagrangian at a point. We also show how suitable underestimators can be constructed for the Lagrangian in the case where the problem has bounds on the variables. Examples are given to illustrate our results.     

Solution sensitivity for Karush–Kuhn–Tucker systems with non-unique Lagrange multipliers

This article is devoted to quantitative stability of a given primal-dual solution of the Karush–Kuhn–Tucker system subject to parametric perturbations. We are mainly concerned with those cases when the dual solution associated with the base primal solution is non-unique. Starting with a review of known results regarding the Lipschitz-stable case, supplied by simple direct justifications based on piecewise analysis, we then proceed with new results for the cases of Hölder (square root) stability. Our results include characterizations of asymptotic behaviour and upper estimates of perturbed solutions, as well as some sufficient conditions for (the specific kinds of) stability of a given solution subject to directional perturbations. We argue that Lipschitz stability of strictly complementary multipliers is highly unlikely to occur, and we employ the recently introduced notion of a critical multiplier for dealing with Hölder stability.     

A simple and elementary proof of the Karush–Kuhn–Tucker theorem for inequality-constrained optimization

We present an elementary proof of the Karush–Kuhn–Tucker Theorem for the problem with nonlinear inequality constraints and linear equality constraints. Most proofs in the literature rely on advanced optimization concepts such as linear programming duality, the convex separation theorem, or a theorem of the alternative for systems of linear inequalities. By contrast, the proof given here uses only basic facts from linear algebra and the definition of differentiability.     

Karush–Kuhn–Tucker Conditions in Set Optimization

This paper investigates set optimization problems in finite dimensional spaces with the property that the images of the set-valued objective map are described by inequalities and equalities and that sets are compared with the set less order relation. For these problems new Karush–Kuhn–Tucker conditions are shown as necessary and sufficient optimality conditions. Optimality conditions without multiplier of the objective map are also presented. The usefulness of these results is demonstrated with a standard example.     

Improvement of Karush–Kuhn–Tucker conditions under uncertainties using robust decision making indexes

Almost all practical engineering problems work with uncertainties. Particularly, in chemical engineering problems, uncertainties in process models and measurements increase complexity on optimization modeling. In these cases, points are transformed in regions, and the point conditions need to be extended to its neighborhood. Extension and validation of Karush–Kuhn–Tucker (KKT) conditions under uncertainties scenarios are not trivial. In this paper, we propose two new conditions to improve robustness at second order KKT conditions.     

Enhanced Karush–Kuhn–Tucker condition and weaker constraint qualifications

In this paper we study necessary optimality conditions for nonsmooth optimization problems with equality, inequality and abstract set constraints. We derive the enhanced Fritz John condition which contains some new information even in the smooth case than the classical enhanced Fritz John condition. From this enhanced Fritz John condition we derive the enhanced Karush–Kuhn–Tucker condition and introduce the associated pseudonormality and quasinormality condition. We prove that either pseudonormality or quasinormality with regularity on the constraint functions and the set constraint implies the existence of a local error bound. Finally we give a tighter upper estimate for the Fréchet subdifferential and the limiting subdifferential of the value function in terms of quasinormal multipliers which is usually a smaller set than the set of classical normal multipliers. In particular we show that the value function of a perturbed problem is Lipschitz continuous under the perturbed quasinormality condition which is much weaker than the classical normality condition.     

Properties of equation reformulation of the Karush–Kuhn–Tucker condition for nonlinear second order cone optimization problems

We give an equation reformulation of the Karush–Kuhn–Tucker (KKT) condition for the second order cone optimization problem. The equation is strongly semismooth and its Clarke subdifferential at the KKT point is proved to be nonsingular under the constraint nondegeneracy condition and a strong second order sufficient optimality condition. This property is used in an implicit function theorem of semismooth functions to analyze the convergence properties of a local sequential quadratic programming type (for short, SQP-type) method by Kato and Fukushima (Optim Lett 1:129–144, 2007). Moreover, we prove that, a local solution x* to the second order cone optimization problem is a strict minimizer of the Han penalty merit function when the constraint nondegeneracy condition and the strong second order optimality condition are satisfied at x*.     

Strong Kuhn–Tucker conditions and constraint qualifications in locally Lipschitz multiobjective optimization problems

We consider a Pareto multiobjective optimization problem with a feasible set defined by inequality and equality constraints and a set constraint, where the objective and inequality constraints are locally Lipschitz, and the equality constraints are Fréchet differentiable. We study several constraint qualifications in the line of Maeda (J. Optim. Theory Appl. 80: 483–500, 1994) and, under the weakest ones, we establish strong Kuhn–Tucker necessary optimality conditions in terms of Clarke subdifferentials so that the multipliers of the objective functions are all positive.     

Nonsmooth multiobjective programming: strong Kuhn–Tucker conditions

We consider a multiobjective optimization problem with a feasible set defined by inequality and equality constraints and a set constraint, where the objective and constraint functions are locally Lipschitz. Several constraint qualifications are given in such a way that they generalize the classical ones, when the functions are differentiable. The relationships between them are analyzed. Then, we establish strong Kuhn–Tucker necessary optimality conditions in terms of the Clarke subdifferentials such that the multipliers of the objective function are all positive. Furthermore, sufficient optimality conditions under generalized convexity assumptions are derived. Moreover, the concept of efficiency is used to formulate duality for nonsmooth multiobjective problems. Wolf and Mond–Weir type dual problems are formulated. We also establish the weak and strong duality theorems.     

Sufficient optimality conditions in nondifferentiable optimization ∗

The concept of local cone approximations, introduced in recent years by Ester and Thierfelder, is a powerful tool for establishing optimality conditions. In this paper we show how to use it for obtaining sufficient optimality conditions in nondifferentiable optimization     

When the Karush–Kuhn–Tucker Theorem Fails: Constraint Qualifications and Higher-Order Optimality Conditions for Degenerate Optimization Problems

In this paper, we present higher-order analysis of necessary and sufficient optimality conditions for problems with inequality constraints. The paper addresses the case when the constraints are not assumed to be regular at a solution of the optimization problems. In the first two theorems derived in the paper, we show how Karush–Kuhn–Tucker necessary conditions reduce to a specific form containing the objective function only. Then we present optimality conditions of the Karush–Kuhn–Tucker type in Banach spaces under new regularity assumptions. After that, we analyze problems for which the Karush–Kuhn–Tucker form of optimality conditions does not hold and propose necessary and sufficient conditions for those problems. To formulate the optimality conditions, we introduce constraint qualifications for new classes of nonregular nonlinear optimization. The approach of p-regularity used in the paper can be applied to various degenerate nonlinear optimization problems due to its flexibility and generality.     

An elementary proof of the Karush–Kuhn–Tucker theorem in normed linear spaces for problems with a finite number of inequality constraints

We present an elementary proof of the Karush–Kuhn–Tucker theorem for the problem with a finite number of nonlinear inequality constraints in normed linear spaces under the linear independence constraint qualification. Most proofs in the literature rely on advanced concepts and results such as the convex separation theorem and Farkas, lemma. By contrast, the proofs given in this article, including a proof of the lemma, employ only basic results from linear algebra. The lemma derived in this article represents an independent theoretical result.