Abstract: This paper concerns saddle points of rational functions, under general constraints. Based on optimality conditions, we propose an algorithm for computing saddle points. It uses Lasserre’s hierarchy of semidefinite relaxation. The algorithm can get a saddle point if it exists, or it can detect its nonexistence if it does not. Numerical experiments show that the algorithm is efficient for computing saddle points of rational functions. PubDate: 2020-04-01

Abstract: This paper is concerned with the stochastic structured tensors to stochastic complementarity problems. The definitions and properties of stochastic structured tensors, such as the stochastic strong P-tensors, stochastic P-tensors, stochastic \(P_{0}\)-tensors, stochastic strictly semi-positive tensors and stochastic S-tensors are given. It is shown that the expected residual minimization formulation (ERM) of the stochastic structured tensor complementarity problem has a nonempty and bounded solution set. Interestingly, we partially answer the open questions proposed by Che et al. (Optim Lett 13:261–279, 2019). We also consider the expected value method of stochastic structured tensor complementarity problem with finitely many elements probability space. Finally, based on the expected residual minimization formulation (ERM) of the stochastic structured tensor complementarity problem, a projected gradient method is proposed for solving the stochastic structured tensor complementarity problem and the related numerical results are also given to show the efficiency of the proposed method. PubDate: 2020-04-01

Abstract: In this paper, we study the higher-degree tensor eigenvalue complementarity problem (HDTEiCP). We give an upper bound for the number of the higher-degree complementarity eigenvalues for the generic HDTEiCP. A semidefinite relaxation algorithm is proposed for computing all the higher-degree complementarity eigenvalues sequentially, as well as the corresponding eigenvectors, and the convergence of the algorithm is discussed. Some numerical results are also given. PubDate: 2020-04-01

Abstract: The purpose of this paper is to study the problem of computing unitary eigenvalues (U-eigenvalues) of non-symmetric complex tensors. By means of symmetric embedding of complex tensors, the relationship between U-eigenpairs of a non-symmetric complex tensor and unitary symmetric eigenpairs (US-eigenpairs) of its symmetric embedding tensor is established. An Algorithm 3.1 is given to compute the U-eigenvalues of non-symmetric complex tensors by means of symmetric embedding. Another Algorithm 3.2, is proposed to directly compute the U-eigenvalues of non-symmetric complex tensors, without the aid of symmetric embedding. Finally, a tensor version of the well-known Gauss–Seidel method is developed. Efficiency of these three algorithms are compared by means of various numerical examples. These algorithms are applied to compute the geometric measure of entanglement of quantum multipartite non-symmetric pure states. PubDate: 2020-04-01

Abstract: \(\mathbf {P}\)-tensor and \(\mathbf {P}_0\)-tensor are introduced in tensor complementarity problem, which have wide applications in game theory. In this paper, we establish SDP relaxation algorithms for detecting \(\mathbf {P}(\mathbf {P}_0)\)-tensor. We first reformulate \(\mathbf {P}(\mathbf {P}_0)\)-tensor detection problem as polynomial optimization problems. Then we propose the SDP relaxation algorithms for solving the reformulated polynomial optimization problems. Numerical examples are reported to show the efficiency of the proposed algorithms. PubDate: 2020-04-01

Abstract: Tensor decompositions have become increasingly prevalent in recent years. Traditionally, tensors are represented or decomposed as a sum of rank-one outer products using either the CANDECOMP/PARAFAC, the Tucker model, or some variations thereof. The motivation of these decompositions is to find an approximate representation for a given tensor. The main propose of this paper is to develop two neural network models for finding an approximation based on t-product for a given third-order tensor. Theoretical analysis shows that each of the neural network models ensures the convergence performance. The computer simulation results further substantiate that the models can find effectively the left and right singular tensor subspace. PubDate: 2020-04-01

Abstract: This paper discusses how to compute all real solutions of the second-order cone tensor complementarity problem when there are finitely many ones. For this goal, we first formulate the second-order cone tensor complementarity problem as two polynomial optimization problems. Based on the reformulation, a semidefinite relaxation method is proposed by solving a finite number of semidefinite relaxations with some assumptions. Numerical experiments are given to show the efficiency of the method. PubDate: 2020-04-01

Abstract: A strongly orthogonal decomposition of a tensor is a rank one tensor decomposition with the two component vectors in each mode of any two rank one tensors are either colinear or orthogonal. A strongly orthogonal decomposition with few number of rank one tensors is favorable in applications, which can be represented by a matrix-tensor multiplication with orthogonal factor matrices and a sparse tensor; and such a decomposition with the minimum number of rank one tensors is a strongly orthogonal rank decomposition. Any tensor has a strongly orthogonal rank decomposition. In this article, computing a strongly orthogonal rank decomposition is equivalently reformulated as solving an optimization problem. Different from the ill-posedness of the usual optimization reformulation for the tensor rank decomposition problem, the optimization reformulation of the strongly orthogonal rank decomposition of a tensor is well-posed. Each feasible solution of the optimization problem gives a strongly orthogonal decomposition of the tensor; and a global optimizer gives a strongly orthogonal rank decomposition, which is however difficult to compute. An inexact augmented Lagrangian method is proposed to solve the optimization problem. The augmented Lagrangian subproblem is solved by a proximal alternating minimization method, with the advantage that each subproblem has a closed formula solution and the factor matrices are kept orthogonal during the iteration. Thus, the algorithm always can return a feasible solution and thus a strongly orthogonal decomposition for any given tensor. Global convergence of this algorithm to a critical point is established without any further assumption. Extensive numerical experiments are conducted, and show that the proposed algorithm is quite promising in both efficiency and accuracy. PubDate: 2020-04-01

Abstract: We consider the semi-infinite system of polynomial inequalities of the form $$\begin{aligned} {{\mathbf {K}}}:=\{x\in {{\mathbb {R}}}^m\mid p(x,y)\ge 0,\quad \forall y\in S\subseteq {{\mathbb {R}}}^n\}, \end{aligned}$$where p(x, y) is a real polynomial in the variables x and the parameters y, the index set S is a basic semialgebraic set in \({{\mathbb {R}}}^n\), \(-p(x,y)\) is convex in x for every \(y\in S\). We propose a procedure to construct approximate semidefinite representations of \({{\mathbf {K}}}\). There are two indices to index these approximate semidefinite representations. As two indices increase, these semidefinite representation sets expand and contract, respectively, and can approximate \({{\mathbf {K}}}\) as closely as possible under some assumptions. In some special cases, we can fix one of the two indices or both. Then, we consider the optimization problem of minimizing a convex polynomial over \({{\mathbf {K}}}\). We present an SDP relaxation method for this optimization problem by similar strategies used in constructing approximate semidefinite representations of \({{\mathbf {K}}}\). Under certain assumptions, some approximate minimizers of the optimization problem can also be obtained from the SDP relaxations. In some special cases, we show that the SDP relaxation for the optimization problem is exact and all minimizers can be extracted. PubDate: 2020-04-01

Abstract: This paper considers a class of two-stage stochastic linear variational inequality problems whose first stage problems are stochastic linear box-constrained variational inequality problems and second stage problems are stochastic linear complementary problems having a unique solution. We first give conditions for the existence of solutions to both the original problem and its perturbed problems. Next we derive quantitative stability assertions of this two-stage stochastic problem under total variation metrics via the corresponding residual function. Moreover, we study the discrete approximation problem. The convergence and the exponential rate of convergence of optimal solution sets are obtained under moderate assumptions respectively. Finally, through solving a non-cooperative game in which each player’s problem is a parameterized two-stage stochastic program, we numerically illustrate our theoretical results. PubDate: 2020-03-21

Abstract: Nonlinear disjunctive convex sets arise naturally in the formulation or solution methods of many discrete–continuous optimization problems. Often, a tight algebraic representation of the disjunctive convex set is sought, with the tightest such representation involving the characterization of the convex hull of the disjunctive convex set. In the most general case, this can be explicitly expressed through the use of the perspective function in higher dimensional space—the so-called extended formulation of the convex hull of a disjunctive convex set. However, there are a number of challenges in using this characterization in computation which prevents its wide-spread use, including issues that arise because of the functional form of the perspective function. In this paper, we propose an explicit algebraic representation of a fairly large class of nonlinear disjunctive convex sets using the perspective function that addresses this latter computational challenge. This explicit representation can be used to generate (tighter) algebraic reformulations for a variety of different problems containing disjunctive convex sets, and we report illustrative computational results using this representation for several nonlinear disjunctive problems. PubDate: 2020-03-12

Abstract: This paper studies an acceleration technique for incremental aggregated gradient (IAG) method through the use of curvature information for solving strongly convex finite sum optimization problems. These optimization problems of interest arise in large-scale learning applications. Our technique utilizes a curvature-aided gradient tracking step to produce accurate gradient estimates incrementally using Hessian information. We propose and analyze two methods utilizing the new technique, the curvature-aided IAG (CIAG) method and the accelerated CIAG (A-CIAG) method, which are analogous to gradient method and Nesterov’s accelerated gradient method, respectively. Setting \(\kappa\) to be the condition number of the objective function, we prove the R linear convergence rates of \(1 - \frac{4c_0 \kappa }{(\kappa +1)^2}\) for the CIAG method, and \(1 - \sqrt{\frac{c_1}{2\kappa }}\) for the A-CIAG method, where \(c_0,c_1 \le 1\) are constants inversely proportional to the distance between the initial point and the optimal solution. When the initial iterate is close to the optimal solution, the R linear convergence rates match with the gradient and accelerated gradient method, albeit CIAG and A-CIAG operate in an incremental setting with strictly lower computation complexity. Numerical experiments confirm our findings. The source codes used for this paper can be found on http://github.com/hoitowai/ciag/. PubDate: 2020-03-07

Abstract: The multi-objective spanning tree (MoST) is an extension of the minimum spanning tree problem (MST) that, as well as its single-objective counterpart, arises in several practical applications. However, unlike the MST, for which there are polynomial-time algorithms that solve it, the MoST is NP-hard. Several researchers proposed techniques to solve the MoST, each of those methods with specific potentialities and limitations. In this study, we examine those methods and divide them into two categories regarding their outcomes: Pareto optimal sets and Pareto optimal fronts. To compare the techniques from the two groups, we investigated their behavior on 2, 3 and 4-objective instances from different classes. We report the results of a computational experiment on 8100 complete and grid graphs in which we analyze specific features of each algorithm as well as the computational effort required to solve the instances. PubDate: 2020-03-01

Abstract: Many real-world applications can usually be modeled as convex quadratic problems. In the present paper, we want to tackle a specific class of quadratic programs having a dense Hessian matrix and a structured feasible set. We hence carefully analyze a simplicial decomposition like algorithmic framework that handles those problems in an effective way. We introduce a new master solver, called Adaptive Conjugate Direction Method, and embed it in our framework. We also analyze the interaction of some techniques for speeding up the solution of the pricing problem. We report extensive numerical experiments based on a benchmark of almost 1400 instances from specific and generic quadratic problems. We show the efficiency and robustness of the method when compared to a commercial solver (Cplex). PubDate: 2020-03-01

Abstract: In this paper, we introduce and analyze a new algorithm for solving equilibrium problem involving pseudomonotone and Lipschitz-type bifunction in real Hilbert space. The algorithm requires only a strongly convex programming problem per iteration. A weak and a strong convergence theorem are established without the knowledge of the Lipschitz-type constants of the bifunction. As a special case of equilibrium problem, the variational inequality is also considered. Finally, numerical experiments are performed to illustrate the advantage of the proposed algorithm. PubDate: 2020-03-01

Abstract: The alternating direction method of multipliers (ADMM) is being widely used in a variety of areas; its different variants tailored for different application scenarios have also been deeply researched in the literature. Among them, the linearized ADMM has received particularly wide attention in many areas because of its efficiency and easy implementation. To theoretically guarantee convergence of the linearized ADMM, the step size for the linearized subproblems, or the reciprocal of the linearization parameter, should be sufficiently small. On the other hand, small step sizes decelerate the convergence numerically. Hence, it is interesting to probe the optimal (largest) value of the step size that guarantees convergence of the linearized ADMM. This analysis is lacked in the literature. In this paper, we provide a rigorous mathematical analysis for finding this optimal step size of the linearized ADMM and accordingly set up the optimal version of the linearized ADMM in the convex programming context. The global convergence and worst-case convergence rate measured by the iteration complexity of the optimal version of linearized ADMM are proved as well. PubDate: 2020-03-01

Abstract: This paper presents a generalization of the spectral norm and the nuclear norm of a tensor via arbitrary tensor partitions, a much richer concept than block tensors. We show that the spectral p-norm and the nuclear p-norm of a tensor can be lower and upper bounded by manipulating the spectral p-norms and the nuclear p-norms of subtensors in an arbitrary partition of the tensor for \(1\le p\le \infty\). Hence, it generalizes and answers affirmatively the conjecture proposed by Li (SIAM J Matrix Anal Appl 37:1440–1452, 2016) for a tensor partition and \(p=2\). We study the relations of the norms of a tensor, the norms of matrix unfoldings of the tensor, and the bounds via the norms of matrix slices of the tensor. Various bounds of the tensor spectral and nuclear norms in the literature are implied by our results. PubDate: 2020-02-20

Abstract: We study a facility location problem where a single facility serves multiple customers each represented by a (possibly non-convex) region in the plane. The aim of the problem is to locate a single facility in the plane so that the maximum of the closest Euclidean distances between the facility and the customer regions is minimized. Assuming that each customer region is mixed-integer second order cone representable, we firstly give a mixed-integer second order cone programming formulation of the problem. Secondly, we consider a solution method based on the Minkowski sums of sets. Both of these solution methods are extended to the constrained case in which the facility is to be located on a (possibly non-convex) subset of the plane. Finally, these two methods are compared in terms of solution quality and time with extensive computational experiments. PubDate: 2020-01-04

Abstract: Quasi-Newton methods are often used in the frame of non-linear optimization. In those methods, the quality and cost of the estimate of the Hessian matrix has a major influence on the efficiency of the optimization algorithm, which has a huge impact for computationally costly problems. One strategy to create a more accurate estimate of the Hessian consists in maximizing the use of available information during this computation. This is done by combining different characteristics. The Powell-Symmetric-Broyden method (PSB) imposes, for example, the satisfaction of the last secant equation, which is called secant update property, and the symmetry of the Hessian (Powell in Nonlinear Programming 31–65, 1970). Imposing the satisfaction of more secant equations should be the next step to include more information into the Hessian. However, Schnabel proved that this is impossible (Schnabel in quasi-Newton methods using multiple secant equations, 1983). Penalized PSB (pPSB), works around the impossibility by giving a symmetric Hessian and penalizing the non-satisfaction of the multiple secant equations by using weight factors (Gratton et al. in Optim Methods Softw 30(4):748–755, 2015). Doing so, he loses the secant update property. In this paper, we combine the properties of PSB and pPSB by adding to pPSB the secant update property. This gives us the secant update penalized PSB (SUpPSB). This new formula that we propose also avoids matrix inversions, which makes it easier to compute. Next to that, SUpPSB also performs globally better compared to pPSB. PubDate: 2020-01-02