Tensor eigenvalue complementarity problems

This paper studies tensor eigenvalue complementarity problems. Basic properties of standard and complementarity tensor eigenvalues are discussed. We formulate tensor eigenvalue complementarity problems as constrained polynomial optimization. When one tensor is strictly copositive, the complementarity eigenvalues can be computed by solving polynomial optimization with normalization by strict copositivity. When no tensor is strictly copositive, we formulate the tensor eigenvalue complementarity problem equivalently as polynomial optimization by a randomization process. The complementarity eigenvalues can be computed sequentially. The formulated polynomial optimization can be solved by Lasserre’s hierarchy of semidefinite relaxations. We show that it has finite convergence for generic tensors. Numerical experiments are presented to show the efficiency of proposed methods.     

Completely positive and completely positive semidefinite tensor relaxations for polynomial optimization

Completely positive (CP) tensors, which correspond to a generalization of CP matrices, allow to reformulate or approximate a general polynomial optimization problem (POP) with a conic optimization problem over the cone of CP tensors. Similarly, completely positive semidefinite (CPSD) tensors, which correspond to a generalization of positive semidefinite (PSD) matrices, can be used to approximate general POPs with a conic optimization problem over the cone of CPSD tensors. In this paper, we study CP and CPSD tensor relaxations for general POPs and compare them with the bounds obtained via a Lagrangian relaxation of the POPs. This shows that existing results in this direction for quadratic POPs extend to general POPs. Also, we provide some tractable approximation strategies for CP and CPSD tensor relaxations. These approximation strategies show that, with a similar computational effort, bounds obtained from them for general POPs can be tighter than bounds for these problems obtained by reformulating the POP as a quadratic POP, which subsequently can be approximated using CP and PSD matrices. To illustrate our results, we numerically compare the bounds obtained from these relaxation approaches on small scale fourth-order degree POPs.     

Convex relaxations for nonconvex quadratically constrained quadratic programming: matrix cone decomposition and polyhedral approximation

We present a decomposition-approximation method for generating convex relaxations for nonconvex quadratically constrained quadratic programming (QCQP). We first develop a general conic program relaxation for QCQP based on a matrix decomposition scheme and polyhedral (piecewise linear) underestimation. By employing suitable matrix cones, we then show that the convex conic relaxation can be reduced to a semidefinite programming (SDP) problem. In particular, we investigate polyhedral underestimations for several classes of matrix cones, including the cones of rank-1 and rank-2 matrices, the cone generated by the coefficient matrices, the cone of positive semidefinite matrices and the cones induced by rank-2 semidefinite inequalities. We demonstrate that in general the new SDP relaxations can generate lower bounds at least as tight as the best known SDP relaxations for QCQP. Moreover, we give examples for which tighter lower bounds can be generated by the new SDP relaxations. We also report comparison results of different convex relaxation schemes for nonconvex QCQP with convex quadratic/linear constraints, nonconvex quadratic constraints and 0–1 constraints.     

Approximating Pareto curves using semidefinite relaxations

We approximate as closely as desired the Pareto curve associated with bicriteria polynomial optimization problems. We use three formulations (including the weighted sum approach and the Chebyshev approximation) and each of them is viewed as a parametric polynomial optimization problem. For each case is associated a hierarchy of semidefinite relaxations and from an optimal solution of each relaxation one approximates the Pareto curve by solving an inverse problem (first two cases) or by building a polynomial underestimator (third case).     

Cuts for mixed 0-1 conic programming

In this we paper we study techniques for generating valid convex constraints for mixed 0-1 conic programs. We show that many of the techniques developed for generating linear cuts for mixed 0-1 linear programs, such as the Gomory cuts, the lift-and-project cuts, and cuts from other hierarchies of tighter relaxations, extend in a straightforward manner to mixed 0-1 conic programs. We also show that simple extensions of these techniques lead to methods for generating convex quadratic cuts. Gomory cuts for mixed 0-1 conic programs have interesting implications for comparing the semidefinite programming and the linear programming relaxations of combinatorial optimization problems, e.g. we show that all the subtour elimination inequalities for the traveling salesman problem are rank-1 Gomory cuts with respect to a single semidefinite constraint. We also include results from our preliminary computational experiments with these cuts.Research partially supported by NSF grants CCR-00-09972, DMS-01-04282 and ONR grant N000140310514.     

Subtracting a best rank-1 approximation may increase tensor rank

It has been shown that a best rank-R approximation of an order-k tensor may not exist when R?2 and k?3. This poses a serious problem to data analysts using tensor decompositions. It has been observed numerically that, generally, this issue cannot be solved by consecutively computing and subtracting best rank-1 approximations. The reason for this is that subtracting a best rank-1 approximation generally does not decrease tensor rank. In this paper, we provide a mathematical treatment of this property for real-valued 2×2×2 tensors, with symmetric tensors as a special case. Regardless of the symmetry, we show that for generic 2×2×2 tensors (which have rank 2 or 3), subtracting a best rank-1 approximation results in a tensor that has rank 3 and lies on the boundary between the rank-2 and rank-3 sets. Hence, for a typical tensor of rank 2, subtracting a best rank-1 approximation increases the tensor rank.     

Conic approximation to quadratic optimization with linear complementarity constraints

This paper proposes a conic approximation algorithm for solving quadratic optimization problems with linear complementarity constraints.We provide a conic reformulation and its dual for the original problem such that these three problems share the same optimal objective value. Moreover, we show that the conic reformulation problem is attainable when the original problem has a nonempty and bounded feasible domain. Since the conic reformulation is in general a hard problem, some conic relaxations are further considered. We offer a condition under which both the semidefinite relaxation and its dual problem become strictly feasible for finding a lower bound in polynomial time. For more general cases, by adaptively refining the outer approximation of the feasible set, we propose a conic approximation algorithm to identify an optimal solution or an \(\epsilon \)-optimal solution of the original problem. A convergence proof is given under simple assumptions. Some computational results are included to illustrate the effectiveness of the proposed algorithm.     

Semidefinite programming for min–max problems and games

We consider two min–max problems (1) minimizing the supremum of finitely many rational functions over a compact basic semi-algebraic set and (2) solving a 2-player zero-sum polynomial game in randomized strategies with compact basic semi-algebraic sets of pure strategies. In both problems the optimal value can be approximated by solving a hierarchy of semidefinite relaxations, in the spirit of the moment approach developed in Lasserre (SIAM J Optim 11:796–817, 2001; Math Program B 112:65–92, 2008). This provides a unified approach and a class of algorithms to compute Nash equilibria and min–max strategies of several static and dynamic games. Each semidefinite relaxation can be solved in time which is polynomial in its input size and practice on a sample of experiments reveals that few relaxations are needed for a good approximation (and sometimes even for finite convergence), a behavior similar to what was observed in polynomial optimization.     

Finding Robust Global Optimal Values of Bilevel Polynomial Programs with Uncertain Linear Constraints

This paper studies a bilevel polynomial program involving box data uncertainties in both its linear constraint set and its lower-level optimization problem. We show that the robust global optimal value of the uncertain bilevel polynomial program is the limit of a sequence of values of Lasserre-type hierarchy of semidefinite linear programming relaxations. This is done by first transforming the uncertain bilevel polynomial program into a single-level non-convex polynomial program using a dual characterization of the solution of the lower-level program and then employing the powerful Putinar’s Positivstellensatz of semi-algebraic geometry. We provide a numerical example to show how the robust global optimal value of the uncertain bilevel polynomial program can be calculated by solving a semidefinite programming problem using the MATLAB toolbox YALMIP.     

A tensor product matrix approximation problem in quantum physics

We consider a matrix approximation problem arising in the study of entanglement in quantum physics. This notion represents a certain type of correlations between subsystems in a composite quantum system. The states of a system are described by a density matrix, which is a positive semidefinite matrix with trace one. The goal is to approximate such a given density matrix by a so-called separable density matrix, and the distance between these matrices gives information about the degree of entanglement in the system. Separability here is expressed in terms of tensor products. We discuss this approximation problem for a composite system with two subsystems and show that it can be written as a convex optimization problem with special structure. We investigate related convex sets, and suggest an algorithm for this approximation problem which exploits the tensor product structure in certain subproblems. Finally some computational results and experiences are presented.     

Generalized Lagrangian duality for nonconvex polynomial programs with polynomial multipliers

In this paper, under the existence of a certificate of nonnegativity of the objective function over the given constraint set, we present saddle-point global optimality conditions and a generalized Lagrangian duality theorem for (not necessarily convex) polynomial optimization problems, where the Lagrange multipliers are polynomials. We show that the nonnegativity certificate together with the archimedean condition guarantees that the values of the Lasserre hierarchy of semidefinite programming (SDP) relaxations of the primal polynomial problem converge asymptotically to the common primal–dual value. We then show that the known regularity conditions that guarantee finite convergence of the Lasserre hierarchy also ensure that the nonnegativity certificate holds and the values of the SDP relaxations converge finitely to the common primal–dual value. Finally, we provide classes of nonconvex polynomial optimization problems for which the Slater condition guarantees the required nonnegativity certificate and the common primal–dual value with constant multipliers and the dual problems can be reformulated as semidefinite programs. These classes include some separable polynomial programs and quadratic optimization problems with quadratic constraints that admit certain hidden convexity. We also give several numerical examples that illustrate our results.     

Sums of squares based approximation algorithms for MAX-SAT

We investigate the Semidefinite Programming based sums of squares (SOS) decomposition method, designed for global optimization of polynomials, in the context of the (Maximum) Satisfiability problem. To be specific, we examine the potential of this theory for providing tests for unsatisfiability and providing MAX-SAT upper bounds. We compare the SOS approach with existing upper bound and rounding techniques for the MAX-2-SAT case of Goemans and Williamson [Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming, J. Assoc. Comput. Mach. 42(6) (1995) 1115-1145] and Feige and Goemans [Approximating the value of two prover proof systems, with applications to MAX2SAT and MAXDICUT, in: Proceedings of the Third Israel Symposium on Theory of Computing and Systems, 1995, pp. 182-189] and the MAX-3-SAT case of Karloff and Zwick [A 7/8-approximation algorithm for MAX 3SAT? in: Proceedings of the 38th Annual IEEE Symposium on Foundations of Computer Science, Miami Beach, FL, USA, IEEE Press, New York, 1997], which are based on Semidefinite Programming as well. We prove that for each of these algorithms there is an SOS-based counterpart which provides upper bounds at least as tight, but observably tighter in particular cases. Also, we propose a new randomized rounding technique based on the optimal solution of the SOS Semidefinite Program (SDP) which we experimentally compare with the appropriate existing rounding techniques. Further we investigate the implications to the decision variant SAT and compare experimental results with those yielded from the higher lifting approach of Anjos [On semidefinite programming relaxations for the satisfiability problem, Math. Methods Oper. Res. 60(3) (2004) 349-367; An improved semidefinite programming relaxation for the satisfiability problem, Math. Programming 102(3) (2005) 589-608; Semidefinite optimization approaches for satisfiability and maximum-satisfiability problems, J. Satisfiability Boolean Modeling Comput. 1 (2005) 1-47].We give some impression of the fraction of the so-called unit constraints in the various SDP relaxations. From a mathematical viewpoint these constraints should be easily dealt within an algorithmic setting, but seem hard to be avoided as extra constraints in an SDP setting. Finally, we briefly indicate whether this work could have implications in finding counterexamples to uncovered cases in Hilbert's Positivstellensatz.     

On linear and semidefinite programming relaxations for hypergraph matching

The hypergraph matching problem is to find a largest collection of disjoint hyperedges in a hypergraph. This is a well-studied problem in combinatorial optimization and graph theory with various applications. The best known approximation algorithms for this problem are all local search algorithms. In this paper we analyze different linear and semidefinite programming relaxations for the hypergraph matching problem, and study their connections to the local search method. Our main results are the following:

We consider the standard linear programming relaxation of the problem. We provide an algorithmic proof of a result of Füredi, Kahn and Seymour, showing that the integrality gap is exactly ${k-1+\frac{1}{k}}$ for k-uniform hypergraphs, and is exactly k ? 1 for k-partite hypergraphs. This yields an improved approximation algorithm for the weighted 3-dimensional matching problem. Our algorithm combines the use of the iterative rounding method and the fractional local ratio method, showing a new way to round linear programming solutions for packing problems.

We study the strengthening of the standard LP relaxation by local constraints. We show that, even after linear number of rounds of the Sherali-Adams lift-and-project procedure on the standard LP relaxation, there are k-uniform hypergraphs with integrality gap at least k ? 2. On the other hand, we prove that for every constant k, there is a strengthening of the standard LP relaxation by only a polynomial number of constraints, with integrality gap at most ${\frac{k+1}{2}}$ for k-uniform hypergraphs. The construction uses a result in extremal combinatorics.

We consider the standard semidefinite programming relaxation of the problem. We prove that the Lovász ${\vartheta}$ -function provides an SDP relaxation with integrality gap at most ${\frac{k+1}{2}}$ . The proof gives an indirect way (not by a rounding algorithm) to bound the ratio between any local optimal solution and any optimal SDP solution. This shows a new connection between local search and linear and semidefinite programming relaxations.

     

Conjugate gradient algorithms for best rank-1 approximation of tensors

Motivated by considerations of pure state entanglement in quantum information, we consider the problem of finding the best rank-1 approximation to an arbitrary r -th order tensor. Reformulating the problem as an optimization problem on the Lie group SU (n₁) ⊗ … ⊗ SU (n_r) of so-called local unitary transformations and exploiting its intrinsic geometry yields a new approach, which finally leads to Riemannian variant of the conjugate gradient algorithm. Numerical simulations support that our method offers an alternative to the higher-order power method for computing the best rank-1 approximation to a tensor. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

张量分析和多项式优化的若干进展

张量分析 (也称多重数值线性代数) 主要包括张量分解和张量特征值的理论和算法,多项式优化主要包括目标和约束均为多项式的一类优化问题的理论和算法. 主要介绍这两个研究领域中若干新的研究结果. 对张量分析部分,主要介绍非负张量H-特征值谱半径的一些性质及求解方法,还介绍非负张量最大 (小) Z-特征值的优化表示及其解法;对多项式优化部分,主要介绍带单位球约束或离散二分单位取值、目标函数为齐次多项式的优化问题及其推广形式的多项式优化问题和半定松弛解法. 最后对所介绍领域的发展趋势做了预测和展望.     

Semidefinite representations for finite varieties

We consider the problem of minimizing a polynomial over a set defined by polynomial equations and inequalities. When the polynomial equations have a finite set of complex solutions, we can reformulate this problem as a semidefinite programming problem. Our semidefinite representation involves combinatorial moment matrices, which are matrices indexed by a basis of the quotient vector space ℝ[x ₁, . . . ,x _n ]/I, where I is the ideal generated by the polynomial equations in the problem. Moreover, we prove the finite convergence of a hierarchy of semidefinite relaxations introduced by Lasserre. Semidefinite approximations can be constructed by considering truncated combinatorial moment matrices; rank conditions are given (in a grid case) that ensure that the approximation solves the original problem to optimality. Supported by the Netherlands Organisation for Scientific Research grant NWO 639.032.203.     

Decomposition-based Method for Sparse Semidefinite Relaxations of Polynomial Optimization Problems

We consider polynomial optimization problems pervaded by a sparsity pattern. It has been shown in Lasserre (SIAM J. Optim. 17(3):822–843, 2006) and Waki et al. (SIAM J. Optim. 17(1):218–248, 2006) that the optimal solution of a polynomial programming problem with structured sparsity can be computed by solving a series of semidefinite relaxations that possess the same kind of sparsity. We aim at solving the former relaxations with a decomposition-based method, which partitions the relaxations according to their sparsity pattern. The decomposition-based method that we propose is an extension to semidefinite programming of the Benders decomposition for linear programs (Benders, Comput. Manag. Sci. 2(1):3–19, 2005).     

Solving polynomial least squares problems via semidefinite programming relaxations

A polynomial optimization problem whose objective function is represented as a sum of positive and even powers of polynomials, called a polynomial least squares problem, is considered. Methods to transform a polynomial least square problem to polynomial semidefinite programs to reduce degrees of the polynomials are discussed. Computational efficiency of solving the original polynomial least squares problem and the transformed polynomial semidefinite programs is compared. Numerical results on selected polynomial least square problems show better computational performance of a transformed polynomial semidefinite program, especially when degrees of the polynomials are larger.     

On Polynomial Optimization Over Non-compact Semi-algebraic Sets

The optimal value of a polynomial optimization over a compact semi-algebraic set can be approximated as closely as desired by solving a hierarchy of semidefinite programs and the convergence is finite generically under the mild assumption that a quadratic module generated by the constraints is Archimedean. We consider a class of polynomial optimization problems with non-compact semi-algebraic feasible sets, for which an associated quadratic module, that is generated in terms of both the objective function and the constraints, is Archimedean. For such problems, we show that the corresponding hierarchy converges and the convergence is finite generically. Moreover, we prove that the Archimedean condition (as well as a sufficient coercivity condition) can be checked numerically by solving a similar hierarchy of semidefinite programs. In other words, under reasonable assumptions, the now standard hierarchy of semidefinite programming relaxations extends to the non-compact case via a suitable modification.     

Approximation algorithms from inexact solutions to semidefinite programming relaxations of combinatorial optimization problems

Semidefinite relaxations of certain combinatorial optimization problems lead to approximation algorithms with performance guarantees. For large-scale problems, it may not be computationally feasible to solve the semidefinite relaxations to optimality. In this paper, we investigate the effect on the performance guarantees of an approximate solution to the semidefinite relaxation for MaxCut, Max2Sat, and Max3Sat. We show that it is possible to make simple modifications to the approximate solutions and obtain performance guarantees that depend linearly on the most negative eigenvalue of the approximate solution, the size of the problem, and the duality gap. In every case, we recover the original performance guarantees in the limit as the solution approaches the optimal solution to the semidefinite relaxation.