arXiv subjects

Dinh Tao Pham

Publications and source records attributed to Dinh Tao Pham.

3 recordsLinked to original sources

arXiv2024

On Difference-of-SOS and Difference-of-Convex-SOS Decompositions for Polynomials

In this article, we are interested in developing polynomial decomposition techniques based on sums-of-squares (SOS), namely the difference-of-sums-of-squares (D-SOS) and the difference-of-convex-sums-of-squares (DC-SOS). In particular, the DC-SOS decomposition is very useful for difference-of-convex (DC) programming formulation of polynomial optimization problems. First, we introduce the cone of convex-sums-of-squares (CSOS) polynomials and discuss its relationship to the sums-of-squares (SOS) polynomials, the non-negative polynomials and the SOS-convex polynomials. Then, we propose the set of D-SOS and DC-SOS polynomials, and prove that any polynomial can be formulated as D-SOS and DC-SOS. The problem of finding D-SOS and DC-SOS decompositions can be formulated as a semi-definite program and solved for any desired precision in polynomial time using interior point methods. Some algebraic properties of CSOS, D-SOS and DC-SOS are established. Second, we focus on establishing several practical algorithms for exact D-SOS and DC-SOS polynomial decompositions without solving any SDP. The numerical performance of the proposed D-SOS and DC-SOS decomposition algorithms and their parallel versions, tested on a dataset of 1750 randomly generated polynomials, is reported.

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arXiv2022

High-order Moment Portfolio Optimization via An Accelerated Difference-of-Convex Programming Approach and Sums-of-Squares

The Mean-Variance-Skewness-Kurtosis (MVSK) portfolio optimization model is a quartic nonconvex polynomial minimization problem over a polytope, which can be formulated as a Difference-of-Convex (DC) program. In this manuscript, we investigate four DC programming approaches for solving the MVSK model. First, two DC formulations based on the projective DC decomposition and the Difference-of-Convex-Sums-of-Squares (DC-SOS) decomposition are established, where the second one is novel. Then, DCA is applied to solve these DC formulations. The convergence analysis of DCA for the MVSK model is established. Second, we propose an accelerated DCA (Boosted-DCA) for solving a general convex constrained DC program involving both smooth and nonsmooth functions. The acceleration is realized by an inexact line search of the Armijo-type along the DC descent direction generated by two consecutive iterates of DCA. The convergence analysis of the Boosted-DCA is established. Numerical simulations of the proposed four DC algorithms on both synthetic and real portfolio datasets are reported. Comparisons with KNITRO, FILTERSD, IPOPT and MATLAB fmincon optimization solvers demonstrate good performance of our methods. Particularly, two DC algorithms with DC-SOS decomposition require less number of iterations, which demonstrates that DC-SOS decomposition can provide better convex over-approximations for polynomials. Moreover, the accelerated versions indeed reduce the number of iterations and achieve the best numerical results.

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arXiv2019

Improved DC Programming Approaches for Solving the Quadratic Eigenvalue Complementarity Problem

In this paper, we discuss the solution of a Quadratic Eigenvalue Complementarity Problem (QEiCP) by using Difference of Convex (DC) programming approaches. We first show that QEiCP can be represented as dc programming problem. Then we investigate different dc programming formulations of QEiCP and discuss their dc algorithms based on a well-known method -- DCA. A new local dc decomposition is proposed which aims at constructing a better dc decomposition regarding to the specific feature of the target problem in some neighborhoods of the iterates. This new procedure yields faster convergence and better precision of the computed solution. Numerical results illustrate the efficiency of the new dc algorithms in practice.

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