First- and second-order optimality conditions for second-order cone and semidefinite programming under a constant rank condition

Abstract

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 by exploiting the eigenvector structure of the problem. In this paper we propose a more general and geometric approach for defining a new extension of this condition to the conic context. The main advantage of our approach is that we are able to recast the strong second-order properties of the constant rank condition in a conic context. In particular, we obtain a second-order necessary optimality condition that is stronger than the classical one obtained under Robinson's constraint qualification, in the sense that it holds for every Lagrange multiplier, even though our condition is independent of Robinson's condition.

Global Convergence of Algorithms Under Constant Rank Conditions for Nonlinear Second-Order Cone Programming

“…Quite surprisingly, after incorporating eigendecompositions into the nondegeneracy condition (and also Robinson's CQ) we obtained a strictly weaker constraint qualification by means of considering only converging sequences of eigenvectors associated with a given point of interest, which was called weak-nondegeneracy (respectively, weak-Robinson's CQ). Moreover, this "sequential approach" allowed us to bypass the main difficulty in generalizing the celebrated constant rank constraint qualification of NLP, to NSDP [9], which is the presence of a potentially non-zero duality gap even in feasible linear problems (see also [12] for a more detailed discussion on this topic). In this paper we bring those concepts to the context of NSOCP where several improvements with respect to the NSDP approach were made.…”

“…Very recently, we have been extending the notions of constant rank-type constraint qualifications to the contexts of NSDP and NSOCP. While [12] follows an implicit function approach pioneered by Janin [29] and giving rise to a definition of CRCQ that enjoys strong second-order properties, in this paper we exploit a sequential approach [7], which allows even weaker conditions to be defined, such as the CPLD condition, while enjoying global convergence properties of several algorithms without assuming boundedness of the set of Lagrange multipliers but still allowing computation of error bounds. Not surprisingly, when extending NLP concepts to the conic context, different points of view may give rise to different possible extensions, each one extending different applications of the concept.…”

“…However, note that any kind of constant rank condition that depends solely on the derivatives of the constraint functions will always be satisfied for every linear problem, implying it cannot be a constraint qualification -see, for instance, [6]. See also [12,Section 2.1] for a detailed discussion on this issue regarding linear problems.…”

“…Recently, we proposed in [12] a new geometrical characterization of CRCQ for NLP using the faces of the non-negative orthant, which was naturally extended to the context of NSOCP as well as nonlinear semidefinite programming (NSDP). This has led us to an alternative constant rank-type constraint qualification that allowed us to derive strong second order optimality conditions for NSDP and NSOCP without assuming compactness of the Lagrange multiplier set, similarly to what is known in NLP [5].…”

“…This has led us to an alternative constant rank-type constraint qualification that allowed us to derive strong second order optimality conditions for NSDP and NSOCP without assuming compactness of the Lagrange multiplier set, similarly to what is known in NLP [5]. However, no application towards algorithms was provided or suggested in [12]. Since the sequential approach from [11] seems more suitable for algorithms, we developed it even further for NSDP problems [9,10] by directly exploiting the eigenvector structure of the problem, overcoming the limitations of the naive approach.…”