A practical second-order optimality condition for cardinality-constrained problems with application to an augmented Lagrangian method

This paper addresses the mathematical programs with cardinality constraints (MPCaC). We first define two new tailored (strong and weak) second-order necessary conditions, MPCaC-SSONC and MPCaC-WSONC. We then propose a constraint qualification (CQ), namely, MPCaC-relaxed constant rank constraint qualification (MPCaC-RCRCQ), and establish the validity of MPCaC-SSONC at minimizers under this new CQ. All the concepts proposed … Read more

On Constraint Qualifications for Second-Order Optimality Conditions Depending on a Single Lagrange Multiplier.

Second-order optimality conditions play an important role in continuous optimization. In this paper, we present and discuss new constraint qualifications to ensure the validity of some well-known second-order optimality conditions. Our main interest is on second-order conditions that can be associated with numerical methods for solving constrained optimization problems. Such conditions depend on a single … Read more

New Constraint Qualifications with Second-Order Properties in Nonlinear Optimization

In this paper we present and discuss new constraint qualifications to ensure the validity of well known second-order properties in nonlinear optimization. Here, we discuss conditions related to the so-called basic second-order condition, where a new notion of polar pairing is introduced in order to replace the polar operation, useful in the first-order case. We … Read more

An extension of Yuan’s Lemma and its applications in optimization

We prove an extension of Yuan’s Lemma to more than two matrices, as long as the set of matrices has rank at most 2. This is used to generalize the main result of [A. Baccari and A. Trad. On the classical necessary second-order optimality conditions in the presence of equality and inequality constraints. SIAM J. … Read more

Some theoretical limitations of second-order algorithms for smooth constrained optimization

In second-order algorithms, we investigate the relevance of the constant rank of the full set of active constraints in ensuring global convergence to a second-order stationary point. We show that second-order stationarity is not expected in the non-constant rank case if the growth of the so-called tangent multipliers, associated with a second-order complementarity measure, is … Read more

A second-order optimality condition with first- and second-order complementarity associated with global convergence of algorithms

We develop a new notion of second-order complementarity with respect to the tangent subspace related to second-order necessary optimality conditions by the introduction of so-called tangent multipliers. We prove that around a local minimizer, a second-order stationarity residual can be driven to zero while controlling the growth of Lagrange multipliers and tangent multipliers, which gives … Read more

On a conjecture in second-order optimality conditions

In this paper we deal with optimality conditions that can be verified by a nonlinear optimization algorithm, where only a single Lagrange multiplier is avaliable. In particular, we deal with a conjecture formulated in [R. Andreani, J.M. Martinez, M.L. Schuverdt, “On second-order optimality conditions for nonlinear programming”, Optimization, 56:529–542, 2007], which states that whenever a … Read more

On the upper Lipschitz property of the KKT mapping for nonlinear semidefinite optimization

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. Article Download View On the upper Lipschitz property of the KKT mapping for nonlinear semidefinite optimization

Characterization of local quadratic growth for strong minima in the optimal control of semi-linear elliptic equations

In this article we consider an optimal control problem of a semi-linear elliptic equation, with bound constraints on the control. Our aim is to characterize local quadratic growth for the cost function J in the sense of strong solutions. This means that the function J growths quadratically over all feasible controls whose associated state is … Read more

Complexity bounds for second-order optimality in unconstrained optimization

This paper examines worst-case evaluation bounds for finding weak minimizers in unconstrained optimization. For the cubic regularization algorithm, Nesterov and Polyak (2006) and Cartis, Gould and Toint (2010) show that at most O(epsilon^{-3}) iterations may have to be performed for finding an iterate which is within epsilon of satisfying second-order optimality conditions. We first show … Read more