A 4-steps elementary proof of existence of Lagrange multipliers
We present a simplified proof of Lagrange’s theorem using only elementary properties of sets and sequences. ArticleDownload View PDF
Daiana O. Santos
Strong global convergence properties of algorithms for nonlinear symmetric cone programming
Sequential optimality conditions have played a major role in proving strong global convergence properties of numerical algorithms for many classes of optimization problems. In particular, the way complementarity is dealt is fundamental to achieve a strong condition. Typically, one uses the inner product structure to measure complementarity, which gives a very general approach to a … Read more
Naive constant rank-type constraint qualifications for multifold second-order cone programming and semidefinite programming
The constant rank constraint qualification, introduced by Janin in 1984 for nonlinear programming, has been extensively used for sensitivity analysis, global convergence of first- and second-order algorithms, and for computing the derivative of the value function. In this paper we discuss naive extensions of constant rank-type constraint qualifications to second-order cone programming and semidefinite programming, … 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
Optimality conditions for nonlinear second-order cone programming and symmetric cone programming
Nonlinear symmetric cone programming (NSCP) generalizes important optimization problems such as nonlinear programming, nonlinear semidefinite programming and nonlinear second-order cone programming (NSOCP). In this work, we present two new optimality conditions for NSCP without constraint qualifications, which implies the Karush-Kuhn-Tucker conditions under a condition weaker than Robinson’s constraint qualification. In addition, we show the relationship … Read more
An Augmented Lagrangian algorithm for nonlinear semidefinite programming applied to the covering problem
In this work we present an Augmented Lagrangian algorithm for nonlinear semidefinite problems (NLSDPs), which is a natural extension of its consolidated counterpart in nonlinear programming. This method works with two levels of constraints; one that is penalized and other that is kept within the subproblems. This is done in order to allow exploiting the … Read more
Optimality conditions and global convergence for nonlinear semidefinite programming
Sequential optimality conditions have played a major role in unifying and extending global convergence results for several classes of algorithms for general nonlinear optimization. In this paper, we extend theses concepts for nonlinear semidefinite programming. We define two sequential optimality conditions for nonlinear semidefinite programming. The first is a natural extension of the so-called Approximate-Karush-Kuhn-Tucker … 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