The most important open problem in Monotone Operator Theory concerns the maximal monotonicity of the sum of two maximally monotone operators provided that the classical Rockafellar’s constraint qualification holds, which is called the “sum problem”. In this paper, we establish the maximal monotonicity of A+ B provided that A and B are maximally monotone operators such that domA ∩ int domB 6= ∅,...

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 (CP...

This paper deals with bilevel programming programs with convex lower level problems. New necessary and sufficient optimality conditions that involve a single-level mathematical program satisfying the linear independence constraint qualification are introduced. These conditions are solved by an interior point technique for nonlinear programming. Neither the optimality conditions nor the algorith...

A generalized Nash equilibrium problem (GNEP) is formulated in which, in addition to pointwise constraints on both the control and state variables, the feasible sets are partially governed by the solutions of a linear elliptic partial differential equation. The decisions (optimal controls) of the players arise in their competitors optimization problems via the righthand side of the partial diff...

A result due to D. Luenberger on the existence of multipliers in a quasiconvex programming problem is extended to the case of constraints given by an arbitrary convex cone under a constraint qualification condition more general than Slater’s condition. The existence of solutions is not assumed. We point out links with even convexity in the sense of Fenchel, quasisubdifferentiability in the sens...

This article deals with a study of the MPEC problem based on a reformulation to a DC problem (Difference of Convex functions). This reformulation is obtained by a partial penalization of the constraints. In this article we prove that a classical optimality condition for a DC program, if a constraint qualification is satisfied for MPEC, it is a necessary and sufficient condition for a feasible p...

We analyze the perturbations of quasi-solutions to a parameterized nonlinear programming problem, these being feasible solutions accompanied by a Lagrange multiplier vector such that the Karush-Kuhn-Tucker optimality conditions are satisfied. We show under a standard constraint qualification, not requiring uniqueness of the multipliers, that the quasi-solution mapping is differentiable in a gen...

The paper is concerning about the basic optimization problem of projecting a point onto a convex set. We present a class of methods where the problem is reduced to a sequence of projections onto the intersection of several balls. The subproblems are much simpler and more tractable, but the main advantage is that, in so doing, we can avoid solving linear systems completely and thus the methods a...

Necessary conditions in terms of a local minimum principle are derived for optimal control problems subject to index-1 differential-algebraic equations, pure state constraints and mixed control-state constraints. Differential-algebraic equations are composite systems of differential equations and algebraic equations, which frequently arise in practical applications. The local minimum principle ...

The phrase convex optimization refers to the minimization of a convex function over a convex set. However the feasible convex set need not be always described by convex inequalities. In this article we consider a convex feasible set which are described by inequality constraints which are locally Lipschitz and not necessarily convex and need not be smooth. We show that if the Slater’s constraint...