The Lagrange dual function is: g(u, v) = min x L(x, u, v) The corresponding dual problem is: maxu,v g(u, v) subject to u ≥ 0 The Lagrange dual function can be viewd as a pointwise maximization of some affine functions so it is always concave. The dual problem is always convex even if the primal problem is not convex. For any primal problem and dual problem, the weak duality always holds: f∗ ≥ g...

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 to cast KKT systems as a minimization problem with nonnegativity constraints on some of the variables. We prove that, under fairly mild assumpti...

Approximate optimality conditions for a class of nonconvex semi-infinite programs involving support functions are given. The objective function and the constraint functions are locally Lipschitz functions on n . By using a Karush-Kuhn-Tucker KKT condition, we deduce a necessary optimality condition for local approximate solutions. Then, generalized KKT conditions for the problems are proposed. ...

The Karush–Kuhn–Tucker (KKT) optimality conditions and saddle point optimality conditions in fuzzy programming problems have been studied in literature by various authors under different conditions. In this paper, by considering a partial order relation on the set of fuzzy numbers, and convexity with differentiability of fuzzy mappings, we have obtained the Fritz John (FJ) constraint qualificat...

We consider the solution of generalized Nash equilibrium problems by concate-nating the KKT optimality conditions of each player’s optimization problem into a singleKKT-like system. We then propose two approaches for solving this KKT system. The firstapproach is rather simple and uses a merit-function/equation-based technique for the solutionof the KKT system. The second approac...

We consider the solution of generalized Nash equilibrium problems by concate-nating the KKT optimality conditions of each player’s optimization problem into a singleKKT-like system. We then propose two approaches for solving this KKT system. The firstapproach is rather simple and uses a merit-function/equation-based technique for the solutionof the KKT system. The second approac...

This paper presents a voltage stability constrained optimal power flow that is expressed via a bilevel programming framework. The inner objective function is dedicated for maximizing voltage stability margin while the outer objective function is focused on minimization of total production cost of thermal units. The original two stage problem is converted to a single level optimization problem v...

In this note, we prove that the KKT mapping for nonlinear semidefinite optimization problem is upper Lipschitz continuous at the KKT point, under the second-order sufficient optimality conditions and the strict Robinson constraint qualification.

In this paper we consider the linear symmetric cone programming (SCP). At a KarushKuhn-Tucker (KKT) point of SCP, we present the important equivalent conditions for the nonsingularity of Clarke’s generalized Jacobian of the KKT nonsmooth system, such as primal and dual constraint nondegeneracy, the strong regularity, and the nonsingularity of the B-subdifferential of the KKT system. This affirm...

Variational inequalities over sets defined by systems of equalities and inequalities are considered. A new reformulation of the KKT-conditions of the variational inequality as a system of equations is proposed. A related unconstrained minimization reformulation is also investigated. As a by-product of the analysis, a new characterization of strong regularity of KKT-points is given.