arXiv subjects

N. H. Chieu

Publications and source records attributed to N. H. Chieu.

3 recordsLinked to original sources

arXiv2018

Convexifiability of Continuous and Discrete Nonnegative Quadratic Programs for Gap-Free Duality

In this paper we show that a convexifiability property of nonconvex quadratic programs with nonnegative variables and quadratic constraints guarantees zero duality gap between the quadratic programs and their semi-Lagrangian duals. More importantly, we establish that this convexifiability is hidden in classes of nonnegative homogeneous quadratic programs and discrete quadratic programs, such as mixed integer quadratic programs, revealing zero duality gaps. As an application, we prove that robust counterparts of uncertain mixed integer quadratic programs with objective data uncertainty enjoy zero duality gaps under suitable conditions. Various sufficient conditions for convexifiability are also given.

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arXiv2017

SOS-convex Semi-algebraic Programs and its Applications to Robust Optimization: A Tractable Class of Nonsmooth Convex Optimization

In this paper, we introduce a new class of nonsmooth convex functions called SOS-convex semialgebraic functions extending the recently proposed notion of SOS-convex polynomials. This class of nonsmooth convex functions covers many common nonsmooth functions arising in the applications such as the Euclidean norm, the maximum eigenvalue function and the least squares functions with $\ell_1$-regularization or elastic net regularization used in statistics and compressed sensing. We show that, under commonly used strict feasibility conditions, the optimal value and an optimal solution of SOS-convex semi-algebraic programs can be found by solving a single semi-definite programming problem (SDP). We achieve the results by using tools from semi-algebraic geometry, convex-concave minimax theorem and a recently established Jensen inequality type result for SOS-convex polynomials. As an application, we outline how the derived results can be applied to show that robust SOS-convex optimization problems under restricted spectrahedron data uncertainty enjoy exact SDP relaxations. This extends the existing exact SDP relaxation result for restricted ellipsoidal data uncertainty and answers the open questions left in [Optimization Letters 9, 1-18(2015)] on how to recover a robust solution from the semi-definite programming relaxation in this broader setting.

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arXiv2015

Coderivative characterizations of maximal monotonicity for set-valued mappings

This paper concerns generalized differential characterizations of maximal monotone set-valued mappings. Using advanced tools of variational analysis, we establish coderivative criteria for maximal monotonicity of set-valued mappings, which seem to be the first infinitesimal characterizations of maximal monotonicity outside the single-valued case. We also present second-order necessary and sufficient conditions for lower-${\mathcal C}^2$ functions to be convex and strongly convex. Examples are provided to illustrate the obtained results and the imposed assumptions.

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