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.     

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.     

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.     

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.     

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.     

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*.     

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.     

Higher-order Karush–Kuhn–Tucker optimality conditions for set-valued optimization with nonsolid ordering cones

In this paper, we consider higher-order Karush–Kuhn–Tucker optimality conditions in terms of radial derivatives for set-valued optimization with nonsolid ordering cones. First, we develop sum rules and chain rules in the form of equality for radial derivatives. Then, we investigate set-valued optimization including mixed constraints with both ordering cones in the objective and constraint spaces having possibly empty interior. We obtain necessary conditions for quasi-relative efficient solutions and sufficient conditions for Pareto efficient solutions. For the special case of weak efficient solutions, we receive even necessary and sufficient conditions. Our results are new or improve recent existing ones in the literature.     

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.     

Pareto Optimizing and Kuhn–Tucker Stationary Sequences

We study the relation between weakly Pareto minimizing and Kuhn–Tucker stationary nonfeasible sequences for vector optimization under constraints, where the weakly Pareto (efficient) set may be empty. The work is placed in a context of Banach spaces and the constraints are described by a functional taking values in a cone. We characterize the asymptotic feasibility in terms of the constraint map and the asymptotic efficiency via a Kuhn–Tucker system completely approximate, distinguishing the classical bounded case from the nontrivial unbounded one. The latter requires Auslender–Crouzeix type conditions and Ekeland's variational principle for constrained vector problems.     

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.     

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.     

Solving the Karush–Kuhn–Tucker system of a nonconvex programming problem on an unbounded set

In the papers [G.C. Feng, B. Yu, Combined homotopy interior point method for nonlinear programming problems, in: H. Fujita, M. Yamaguti (Eds.), Advances in Numerical Mathematics; Proceedings of the Second Japan–China Seminar on Numerical Mathematics, in: Lecture Notes in Numerical and Applied Analysis, vol. 14, Kinokuniya, Tokyo, 1995, pp. 9–16; G.C. Feng, Z.H. Lin, B. Yu, Existence of an interior pathway to a Karush–Kuhn–Tucker point of a nonconvex programming problem, Nonlinear Analysis 32 (1998) 761–768; Z.H. Lin, B. Yu, G.C. Feng, A combined homotopy interior point method for convex programming problem, Applied Mathematics and Computation 84 (1997) 193–211], a combined homotopy interior method was presented and global convergence results obtained for nonconvex nonlinear programming when the feasible set is bounded and satisfies the so called normal cone condition. However, for when the feasible set is not bounded, no result has so far been obtained. In this paper, a combined homotopy interior method for nonconvex programming problems on the unbounded feasible set is considered. Under suitable additional assumptions, boundedness of the homotopy path, and hence global convergence, is proven.     

Convexificators and boundedness of the Kuhn–Tucker multipliers set

In this paper we consider a nonsmooth optimization problem with equality, inequality and set constraints. We propose new constraint qualifications and Kuhn–Tucker type necessary optimality conditions for this problem involving locally Lipschitz functions. The main tool of our approach is the notion of convexificators. We introduce a nonsmooth version of the Mangasarian–Fromovitz constraint qualification and show that this constraint qualification is necessary and sufficient for the Kuhn–Tucker multipliers set to be nonempty and bounded.     

New perspective on the Kuhn–Tucker theorem

A new proof of the Kuhn–Tucker theorem on necessary conditions for a minimum of a differentiable function of several variables in the case of inequality constraints is given. The proof relies on a simple inequality (common in textbooks) for the projection of a vector onto a convex set.