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

Ellen H. Fukuda (ellen***at***i.kyoto-u.ac.jp)
Gabriel Haeser (ghaeser***at***ime.usp.br)
Leonardo M. Mito (leokoto***at***ime.usp.br)

Abstract: 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 Robinsonís constraint qualification and a generalized form of the so-called weak constant rank 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.

Keywords: Optimality conditions, Semidefinite programming, Second-order cone programming

Category 1: Linear, Cone and Semidefinite Programming

Category 2: Linear, Cone and Semidefinite Programming (Second-Order Cone Programming )

Category 3: Linear, Cone and Semidefinite Programming (Semi-definite Programming )


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Entry Submitted: 08/04/2020
Entry Accepted: 08/04/2020
Entry Last Modified: 08/30/2021

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