arXiv:2106.00775v1 [math.OC] 1 Jun 2021 Sequential constant rank constraint qualifications for nonlinear semidefinite and second-order cone programming with applications Roberto Andreani * Gabriel Haeser † Leonardo M. Mito † H´ ector Ram´ ırez C. ‡ June 3, 2021 Abstract We present new constraint qualification conditions for nonlinear semidefinite and second-order cone programming that extend some of the constant rank-type conditions from nonlinear programming. As an application of these conditions, we provide a unified global convergence proof of a class of algorithms to stationary points without assuming neither uniqueness of the Lagrange multiplier nor boundedness of the Lagrange multipliers set. This class of algorithm includes, for instance, general forms of augmented Lagrangian and sequential quadratic programming methods. We also compare these new conditions with some of the existing ones, including the nondegeneracy condition, Robinson’s constraint qualification, and the metric subregularity constraint qualification. Keywords: Constraint qualifications, Semidefinite programming, Second-order cone programming. 1 Introduction Constraint qualification (CQ) conditions play a crucial role in optimization. They permit to establish first- and second-order necessary optimality conditions for local minima and support the convergence theory of many practical algorithms (see, for instance, a unified convergence analysis for a whole class of algorithms by Andreani et al. [10, Thm. 6]). Some of the well-known CQs in nonlinear programming (NLP) are the constant-rank constraint qualification (CRCQ), introduced by Janin [25], and constant positive linear dependence (CPLD) condition. The latter was first conceptualized by Qi and Wei [29], and then proved to be a constraint qualification by Andreani et al. [14]. Moreover, it has been a source of inspiration for other authors to define even weaker constraint qualifications for NLP, such as the constant rank of the subspace component (CRSC) [11], and the relaxed versions of CRCQ [26] and CPLD [10]. Our interest in constant rank-type conditions is motivated, mainly, by their applications towards obtaining global convergence results of iterative algorithms to stationary points without relying on boundedness or uniqueness of Lagrange multipliers. However, several other applications that we do not pursue in this paper may be expected to be extended to the conic context, such as the computation of the derivative of the value function [25, 27] and the validity of strong second-order necessary optimality conditions that do not rely on the whole set of Lagrange multipliers [2]. Besides, their ability of dealing with redundant constraints, up to some extent, gives modellers some degree of freedom without losing regularity or convergence guarantees on algorithms. For instance, the standard NLP trick of replacing one nondegenerate equality constraint by two inequalities of opposite sign does not violate CRCQ, while violating the standard Mangasarian-Fromovitz CQ. As far as we know, no constant-rank type CQ has been proposed in conic programming until recent years. The first extension of CRCQ to nonlinear second-order cone programming (NSOCP) appeared in [35], but it was shown to be incorrect in [3]. A second proposal, presented in [9], consists of transforming some of the conic constraints into NLP constraints via a reduction function, whenever it was possible, and then demanding constant linear dependence of the reduced constraints, locally. This was considered by the authors a naive extension, since it basically avoids the main difficulties that are expected from a conic framework. What both these works have in common is that they somehow neglected the conic structure of the problem. In a recent article [8], we introduced weak notions of regularity for nonlinear semidefinite programming (NSDP) that were defined in terms of the eigenvectors of the constraints – therein called weak-nondegeneracy The authors received financial support from FAPESP (grants 2017/18308-2, 2017/17840-2, and 2018/24293-0), CNPq (grants 301888/2017-5, 303427/2018-3 and 404656/2018-8), and ANID (FONDECYT grant 1201982 and Basal Program CMM ANID PIA AFB170001). * Department of Applied Mathematics, State University of Campinas, Campinas, SP, Brazil. Email: andreani@unicamp.br † Department of Applied Mathematics, University of S˜ ao Paulo, S˜ ao Paulo, SP, Brazil. Emails: ghaeser@ime.usp.br, leokoto@ime.usp.br ‡ Departamento de Ingenier´ ıa Matem´ atica and Centro de Modelamiento Matem´ atico (AFB170001 - CNRS IRL2807), Universidad de Chile, Santiago, Chile. Email: hramirez@dim.uchile.cl 1