Zheng-Hai Huang

Smoothing algorithms for complementarity problems over symmetric cones

There recently has been much interest in studying optimization problems over symmetric cones. In this paper, we first investigate a smoothing function in the context of symmetric cones and show that it is coercive under suitable assumptions. We then extend two generic frameworks of smoothing algorithms to solve the complementarity problems over symmetric cones, and prove the proposed algorithms are globally convergent under suitable assumptions. We also give a specific smoothing Newton algorithm which is globally and locally quadratically convergent under suitable assumptions. The theory of Euclidean Jordan algebras is a basic tool which is extensively used in our analysis. Preliminary numerical results for second-order cone complementarity problems are reported.

Predictor-Corrector Smoothing Newton Method, Based on a New Smoothing Function, for Solving the Nonlinear Complementarity Problem with a P0 Function

By smoothing a perturbed minimum function, we propose in this paper a new smoothing function. The existence and continuity of a smooth path for solving the nonlinear complementarity problem (NCP) with a P0 function are discussed. We investigate the boundedness of the iteration sequence generated by noninterior continuation/smoothing methods under the assumption that the solution set of the NCP is nonempty and bounded. Based on the new smoothing function, we present a predictor-corrector smoothing Newton algorithm for solving the NCP with a P0 function, which is shown to be globally linearly and locally superlinearly convergent under suitable assumptions. Some preliminary computational results are reported.

On determinants and eigenvalue theory of tensors

We generalize the concept of the symmetric hyperdeterminants for symmetric tensors to the E-determinants for general tensors. We show that the E-determinant inherits many properties of the determinant of a matrix. These properties include: solvability of polynomial systems, the E-determinat of the composition of tensors, product formula for the E-determinant of a block tensor, Hadamard’s inequality, Geršgrin’s inequality and Minikowski’s inequality. As a simple application, we show that if the leading coefficient tensor of a polynomial system is a triangular tensor with nonzero diagonal elements, then the system definitely has a solution. We investigate the characteristic polynomial of a tensor through the E-determinant. Explicit formulae for the coefficients of the characteristic polynomial are given when the dimension is two.

Sub-quadratic convergence of a smoothing Newton algorithm for the P 0 – and monotone LCP

Abstract.Given , the linear complementarity problem (LCP) is to find such that (x, s)≥ 0,s=Mx+q,xTs=0. By using the Chen-Harker-Kanzow-Smale (CHKS) smoothing function, the LCP is reformulated as a system of parameterized smooth-nonsmooth equations. As a result, a smoothing Newton algorithm, which is a modified version of the Qi-Sun-Zhou algorithm [Mathematical Programming, Vol. 87, 2000, pp. 1–35], is proposed to solve the LCP with M being assumed to be a P0-matrix (P0–LCP). The proposed algorithm needs only to solve one system of linear equations and to do one line search at each iteration. It is proved in this paper that the proposed algorithm has the following convergence properties: (i) it is well-defined and any accumulation point of the iteration sequence is a solution of the P0–LCP; (ii) it generates a bounded sequence if the P0–LCP has a nonempty and bounded solution set; (iii) if an accumulation point of the iteration sequence satisfies a nonsingularity condition, which implies the P0–LCP has a unique solution, then the whole iteration sequence converges to this accumulation point sub-quadratically with a Q-rate 2–t, where t∈(0,1) is a parameter; and (iv) if M is positive semidefinite and an accumulation point of the iteration sequence satisfies a strict complementarity condition, then the whole sequence converges to the accumulation point quadratically.

Strictly nonnegative tensors and nonnegative tensor partition

In this paper, we introduce a new class of nonnegative tensors — strictly nonnegative tensors. A weakly irreducible nonnegative tensor is a strictly nonnegative tensor but not vice versa. We show that the spectral radius of a strictly nonnegative tensor is always positive. We give some sufficient and necessary conditions for the six well-conditional classes of nonnegative tensors, introduced in the literature, and a full relationship picture about strictly nonnegative tensors with these six classes of nonnegative tensors. We then establish global R-linear convergence of a power method for finding the spectral radius of a nonnegative tensor under the condition of weak irreducibility. We show that for a nonnegative tensor T , there always exists a partition of the index set such that every tensor induced by the partition is weakly irreducible; and the spectral radius of T can be obtained from those spectral radii of the induced tensors. In this way, we develop a convergent algorithm for finding the spectral radius of a general nonnegative tensor without any additional assumption. The preliminary numerical results demonstrate the feasibility and effectiveness of the proposed algorithm.

A note on absolute value equations

In this note, we reformulate a system of absolute value equations (AVEs) as a standard linear complementarity problem (LCP) without any assumption. Utilizing some known results for the LCP, existence and convexity results for the solution set of the AVE are proposed.

Global Uniqueness and Solvability for Tensor Complementarity Problems

Recently, the tensor complementarity problem has been investigated in the literature. An important question involving the property of global uniqueness and solvability for a class of tensor complementarity problems was proposed by Song and Qi (J Optim Theory Appl, 165:854–873, 2015). In the present paper, we give an answer to this question by constructing two counterexamples. We also show that the solution set of this class of tensor complementarity problems is nonempty and compact. In particular, we introduce a class of related structured tensors and show that the corresponding tensor complementarity problem has the property of global uniqueness and solvability.

A Smoothing Newton-Type Algorithm of Stronger Convergence for the Quadratically Constrained Convex Quadratic Programming

In this paper we propose a smoothing Newton-type algorithm for the problem of minimizing a convex quadratic function subject to finitely many convex quadratic inequality constraints. The algorithm is shown to converge globally and possess stronger local superlinear convergence. Preliminary numerical results are also reported.

Formulating an n-person noncooperative game as a tensor complementarity problem

In this paper, we consider a class of n-person noncooperative games, where the utility function of every player is given by a homogeneous polynomial defined by the payoff tensor of that player, which is a natural extension of the bimatrix game where the utility function of every player is given by a quadratic form defined by the payoff matrix of that player. We will call such a problem the multilinear game. We reformulate the multilinear game as a tensor complementarity problem, a generalization of the linear complementarity problem; and show that finding a Nash equilibrium point of the multilinear game is equivalent to finding a solution of the resulted tensor complementarity problem. Especially, we present an explicit relationship between the solutions of the multilinear game and the tensor complementarity problem, which builds a bridge between these two classes of problems. We also apply a smoothing-type algorithm to solve the resulted tensor complementarity problem and give some preliminary numerical results for solving the multilinear games.

Exceptionally regular tensors and tensor complementarity problems

Recently, many structured tensors are defined and their properties are discussed in the literature. In this paper, we introduce a new class of structured tensors, called exceptionally regular (ER) tensor, which is relevant to the tensor complementarity problem (TCP). We show that this class of tensors is a wide class of tensors which includes many important structured tensors as its special cases. By constructing two examples, we demonstrate that an ER-tensor can be, but not always, an R-tensor. We also show that within the class of the semi-positive tensors, the class of ER-tensors coincides with the class of R-tensors. In particular, we consider the TCP with an ER-tensor and show that its solution set is nonempty and compact. In addition, we also obtain that the solution sets of the TCP with an R-tensor or a -tensor are nonempty and compact.