In this work we present new weak conditions that ensure the validity of necessary second order optimality conditions (SOC) for nonlinear optimization. We are able to prove that weak and strong SOCs hold for all Lagrange multipliers using Abadie-type assumptions. We also prove weak and strong SOCs for at least one Lagrange multiplier imposing the Mangasarian-Fromovitz constraint qualification an...

The relationship between the mathematical program with linear complementarity constraints (MPCC) and its inequality relaxation is studied. A new sequential quadratic programming (SQP) method is presented for solving the MPCC based on this relationship. A certain SQP technique is introduced to deal with the possible infeasibility of quadratic programming subproblems. Global convergence results a...

In this paper we discuss second-order properties of the Moreau-Yosida regularization F of a piecewise twice continuously differentiable convex function f . We introduce a new constraint qualification in order to prove that the gradient of F is piecewise continuously differentiable. In addition, we discuss conditions, depending on the Hessians of the pieces, that guarantee positive definiteness ...

We study best approximation problems with nonlinear constraints in Hilbert spaces. The strong “conical hull intersection property” (CHIP) and the “basic constraint qualification” (BCQ) condition are discussed. Best approximations with differentiable constraints and convex constraints are characterized. The analysis generalizes some linearly constrained results of recent works [F. Deutsch, W. Li...

The well known constant rank constraint qualification [Math. Program. Study 21:110–126, 1984] introduced by Janin for nonlinear programming has been recently extended to a conic context exploiting the eigenvector structure of problem. In this paper we propose more general and geometric approach defining new extension condition context. main advantage our is that are able recast strong second-or...

We consider a special class of optimization problems that we call Mathematical Programs with Vanishing Constraints, MPVC for short, which serves as a unified framework for several applications in structural and topology optimization. Since an MPVC most often violates stronger standard constraint qualification, first-order necessary optimality conditions, weaker than the standard KKT-conditions,...

We examine the influence of optimality measures on the benchmarking process, and show that scaling requirements lead to a convergence test for nonlinearly constrained solvers that uses a mixture of absolute and relative error measures. We show that this convergence test is well behaved at any point where the constraints satisfy the Mangasarian-Fromovitz constraint qualification and also avoids ...

Abstrac t . There are well-known first-order sufficient conditions for a point x o to be a strict locally optimal solution of a nonlinear programming problem. In this paper, we show that these conditions also guarantee that Xo is an isolated stationary point of the considered program provided a constraint qualification holds. This result has an interesting application to finite convergence of a...

We investigate ill-posed semidefinite programming problems for which Slater’s constraint qualifications fail, and propose a new reliable termination criterium dealing with such problems. This criterium is scale-independent and provides verified forward error bounds for the true optimal value, where all rounding errors due to floating point arithmetic are taken into account. It is based on a bou...

In this paper, we are concerned with a fractional multiobjective optimization problem ( P ). Using support functions together generalized Guignard constraint qualification, give necessary optimality conditions in terms of convexificators and the Karush–Kuhn–Tucker multipliers. Several intermediate problems have been introduced to help us our investigation.