We present two new constraint qualifications (CQ) that are weaker than the recently introduced Relaxed Constant Positive Linear Depen- dence (RCPLD) constraint qualification. RCPLD is based on the assump- tion that many subsets of the gradients of the active constraints preserve positive linear dependence locally. A major open question was to identify the exact set … Read more
In this paper we consider a mathematical program with semidefinite cone complementarity constraints (SDCMPCC). Such a problem is a matrix analogue of the mathematical program with (vector) complementarity constraints (MPCC) and includes MPCC as a special case. We derive explicit expressions for the strong-, Mordukhovich- and Clarke- (S-, M- and C-)stationary conditions and give constraint … Read more
Augmented Lagrangian methods for derivative-free continuous optimization with constraints are introduced in this paper. The algorithms inherit the convergence results obtained by Andreani, Birgin, Martínez and Schuverdt for the case in which analytic derivatives exist and are available. In particular, feasible limit points satisfy KKT conditions under the Constant Positive Linear Dependence (CPLD) constraint qualification. … Read more
In this work we introduce a relaxed version of the constant positive linear dependence constraint qualification (CPLD) that we call RCPLD. This development is inspired by a recent generalization of the constant rank constraint qualification from Minchenko and Stakhovski that was called RCR. We show that RCPLD is enough to ensure the convergence of an … Read more
In this work we introduce a necessary natural sequential Approximate-Karush-Kuhn-Tucker (AKKT) condition for a point to be a solution of a continuous variational inequality problem without constraint quali cations, and we prove its relation with the Approximate Gradient Projection condition (AGP) of Garciga-Otero and Svaiter. We also prove that a slight variation of the AKKT condition … Read more
Sequential optimality conditions provide adequate theoretical tools to justify stopping criteria for nonlinear programming solvers. Approximate KKT and Approximate Gradient Projection conditions are analyzed in this work. These conditions are not necessarily equivalent. Implications between different conditions and counter-examples will be shown. Algorithmic consequences will be discussed. Article Download View On sequential optimality conditions for … Read more
We discuss two optimal control problems of parabolic equations, with mixed state and control constraints, for which the standard qualification condition does not hold. Our first example is a bottleneck problem, and the second one is an optimal investment problem where a utility type function is to be minimized. By an adapted penalization technique, we … Read more
The Constant Rank condition for feasible points of nonlinear programming problems was defined by Janin in Ref. 1. In that paper the author proved that the condition was a first order constraint qualification. In this work we prove that the Janin Constant Rank condition is, in addition, a second order constraint qualification. We also define … Read more
We present a new relaxation scheme for mathematical programs with equilibrium constraints (MPEC), where the complementarity constraints are replaced by a reformulation that is exact for the complementarity conditions corresponding to sufficiently non-degenerate complementarity components and relaxes only the remaining complementarity conditions. A positive parameter determines to what extent the complementarity conditions are relaxed. The … Read more
The elegant results for strong duality and strict complementarity for linear programming, \LP, can fail for cone programming over nonpolyhedral cones. One can have: unattained optimal values; nonzero duality gaps; and no primal-dual optimal pair that satisfies strict complementarity. This failure is tied to the nonclosure of sums of nonpolyhedral closed cones. We take a … Read more