On Optimality Conditions for Nonlinear Conic Programming

Andreani, Roberto; Gómez, Walter; Haeser, Gabriel; Mito, Leonardo M.; Ramos, Alberto

doi:10.1287/moor.2021.1203

Cited by 15 publications

(10 citation statements)

References 63 publications

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“…Definition 4.1 (Def. 4 of [4]). We say that a point x ∈ F satisfies the AKKT condition when there exist sequences {x k } k∈N → x and {Y k } k∈N ⊆ S m + , and perturbation sequences {δ k } k∈N ⊆ R n and {∆ k } k∈N ⊆ S m , such that:…”

Section: Stronger Sequential-type Constant Rank Cqs For Nsdp and Glob...mentioning

confidence: 97%

Sequential constant rank constraint qualifications for nonlinear semidefinite programming with applications

Andreani,

Haeser,

Mito

et al. 2021

Preprint

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We present new constraint qualification conditions for nonlinear semidefinite 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, sequential quadratic programming, and interior point 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.

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Section: Stronger Sequential-type Constant Rank Cqs For Nsdp and Glob...mentioning

confidence: 97%

Sequential constant rank constraint qualifications for nonlinear semidefinite programming with applications

Andreani,

Haeser,

Mito

et al. 2021

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…An augmented Lagrangian algorithm is also presented in [15] for NSDP, whose global convergence theory is built around AKKT. Such results were sharpened in [5] and further extended in [6], for the general (NCP).…”

Section: Semidefinite Programmingmentioning

confidence: 75%

“…Our approach has a heavy algorithmic taste, as our proof is based on the construction of a sequence of approximate solutions of penalized subproblems, very similarly to a sequence generated by practical algorithms. In particular, a similar first-order approach has recently led to several improvements of global convergence theory of augmented Lagrangian methods in conic contexts [4,5,6,15].…”

Section: Final Remarksmentioning

confidence: 99%

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

Fukuda¹,

Haeser²,

Mito³

2022

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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 weak constant rank-type property which are, together, strictly weaker than nondegeneracy. Our approach is done via a penalty-based strategy, which is aimed at providing strong global convergence results for first-and second-order algorithms. Since we are not assuming strict complementarity, the critical cone does not reduce to a subspace, thus, the second-order condition we arrive at is defined in terms of the lineality space of the critical cone. In the case of nonlinear programming, this condition reduces to the standard second-order condition widely used as second-order stationarity measure in the algorithmic practice.

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“…For any given ε ≥ 0, we say x ∈ IR n is an ε-approximate stationary point for problem (3.1) if there exists a ρ ≥ 0 such that (1). dist(F (x) ; X) ≤ ε, (2). inf w ≤1 dϕ(x)(w) + ρ dp(F (x))(dF (x)w) ≥ −ε where p(y) := dist(y ; X) and dp(y)(w) is the subderivative of p caculated in (2.10).…”

Section: Approximate Stationary Solutionsmentioning

confidence: 99%

Penalty methods to compute stationary solutions in constrained optimization problems

Mohammadi¹

2022

Preprint

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This paper is devoted to studying the stationary solutions of a general constrained optimization problem through its associated unconstrained penalized problems. We aim to answer the question, ' what do the stationary solutions of a penalized unconstrained problem tell us about the stationary solutions of the original constrained optimization problem?". We answer the latter question by establishing relationships between global (local) minimizers and stationary points of the two optimization problems. Given the strong connection between stationary solutions between problems, we introduce a new approximate ε-stationary solution for the constrained optimization problems. We propose an algorithm to compute such an approximate stationary solution for a general constrained optimization problem, even in the absence of Clarke regularity. Under reasonable assumptions, we establish the rate O(ε −2 ) for our algorithm, akin to the gradient descent method for smooth minimization. Since our penalty terms are constructed by the powers of the distance function, our stationarity analysis heavily depends on the generalized differentiation of the distance function. In particular, we characterize the (semi-)differentiability of the distance function dist(.; X) defined by a Fréchet smooth norm, in terms of the geometry of the set X. We show that dist(.; X) is semi-differentiable if and only if X is geometrically derivable. The latter opens the door to design optimization algorithms for constrained optimization problems that suffer from Clarke irregularity in their objectives and constraint functions.

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On Optimality Conditions for Nonlinear Conic Programming

Cited by 15 publications

References 63 publications

Sequential constant rank constraint qualifications for nonlinear semidefinite programming with applications

Sequential constant rank constraint qualifications for nonlinear semidefinite programming with applications

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

Penalty methods to compute stationary solutions in constrained optimization problems

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