Dual solutions in convex stochastic optimization

This paper studies duality and optimality conditions for general convex stochastic optimization problems. The main result gives sufficient conditions for the absence of a duality gap and the existence of dual solutions in a locally convex space of random variables. It implies, in particular, the necessity of scenario-wise optimality conditions that are behind many fundamental … Read more

An active signature method for constrained abs-linear minimization

In this paper we consider the solution of optimization tasks with a piecewise linear objective function and piecewise linear constraints. First, we state optimality conditions for that class of problems using the abs-linearization approach and prove that they can be verified in polynomial time. Subsequently, we propose an algorithm called Constrained Active Signature Method that … Read more

A Unifying Framework for Sparsity Constrained Optimization

In this paper, we consider the optimization problem of minimizing a continuously differentiable function subject to both convex constraints and sparsity constraints. By exploiting a mixed-integer reformulation from the literature, we define a necessary optimality condition based on a tailored neighborhood that allows to take into account potential changes of the support set. We then … Read more

On scaled stopping criteria for a safeguarded augmented Lagrangian method with theoretical guarantees

This paper discusses the use of a stopping criterion based on the scaling of the Karush-Kuhn-Tucker (KKT) conditions by the norm of the approximate Lagrange multiplier in the ALGENCAN implementation of a safeguarded augmented Lagrangian method. Such stopping criterion is already used in several nonlinear programming solvers, but it has not yet been considered in … Read more

On the weak second-order optimality condition for nonlinear semidefinite and second-order cone programming

Second-order necessary optimality conditions for nonlinear conic programming problems that depend on a single Lagrange multiplier are usually built under nondegeneracy and strict complementarity. In this paper we establish a condition of such type for two classes of nonlinear conic problems, namely semidefinite and second-order cone programming, assuming Robinson’s constraint qualification and a generalized form … Read more

On the best achievable quality of limit points of augmented Lagrangian schemes

The optimization literature is vast in papers dealing with improvements on the global convergence of augmented Lagrangian schemes. Usually, the results are based on weak constraint qualifications, or, more recently, on sequential optimality conditions obtained via penalization techniques. In this paper we propose a somewhat different approach, in the sense that the algorithm itself is … Read more

On the use of Jordan Algebras for improving global convergence of an Augmented Lagrangian method in nonlinear semidefinite programming

Jordan Algebras are an important tool for dealing with semidefinite programming and optimization over symmetric cones in general. In this paper, a judicious use of Jordan Algebras in the context of sequential optimality conditions is done in order to generalize the global convergence theory of an Augmented Lagrangian method for nonlinear semidefinite programming. An approximate … Read more

On the convergence of augmented Lagrangian strategies for nonlinear programming

Augmented Lagrangian algorithms are very popular and successful methods for solving constrained optimization problems. Recently, the global convergence analysis of these methods have been dramatically improved by using the notion of the sequential optimality conditions. Such conditions are optimality conditions independently of the fulfilment of any constraint qualifications and provide theoretical tools to justify stopping … Read more

Constraint Qualifications for Karush-Kuhn-Tucker Conditions in Constrained Multiobjective Optimization

The notion of a normal cone of a given set is paramount in optimization and variational analysis. In this work, we give a definition of a multiobjective normal cone which is suitable for studying optimality conditions and constraint qualifications for multiobjective optimization problems. A detailed study of the properties of the multiobjective normal cone is … Read more

On optimality conditions for nonlinear conic programming

Sequential optimality conditions have played a major role in proving stronger global convergence results of numerical algorithms for nonlinear programming. Several extensions have been described in conic contexts, where many open questions have arisen. In this paper, we present new sequential optimality conditions in the context of a general nonlinear conic framework, which explains and … Read more