Theoretical filtering of RLT bound-factor constraints for solving polynomial programming problems to global optimality

In this paper, we propose two sets of theoretically filtered bound-factor constraints for constructing reformulation-linearization technique (RLT)-based linear programming (LP) relaxations for solving polynomial programming problems. We establish related theoretical results for convergence to a global optimum for these reduced sized relaxations, and provide insights into their relative sizes and tightness. Extensive computational results are provided to demonstrate the relative effectiveness of the proposed theoretical filtering strategies in comparison to the standard RLT and a prior heuristic filtering technique using problems from the literature as well as randomly generated test cases.     

Semidefinite relaxations of dynamical programs under discrete constraints

In this work we propose an exact semidefinite relaxation for non-linear, non-convex dynamical programs under discrete constraints in the state variables and the control variables. We outline some theoretical features of the method and workout the solutions of a benchmark problem in cybernetics and the classical inventory problem under discrete constraints.     

Reduced RLT representations for nonconvex polynomial programming problems

This paper explores equivalent, reduced size Reformulation-Linearization Technique (RLT)-based formulations for polynomial programming problems. Utilizing a basis partitioning scheme for an embedded linear equality subsystem, we show that a strict subset of RLT defining equalities imply the remaining ones. Applying this result, we derive significantly reduced RLT representations and develop certain coherent associated branching rules that assure convergence to a global optimum, along with static as well as dynamic basis selection strategies to implement the proposed procedure. In addition, we enhance the RLT relaxations with v-semidefinite cuts, which are empirically shown to further improve the relative performance of the reduced RLT method over the usual RLT approach. We present computational results for randomly generated instances to test the different proposed reduction strategies and to demonstrate the improvement in overall computational effort when such reduced RLT mechanisms are employed.     

Verifying exactness of relaxations for robust semi-definite programs by solving polynomial systems

In this paper, robust semi-definite programs are considered with the goal of verifying whether a particular LMI relaxation is exact. A procedure is presented showing that verifying exactness amounts to solving a polynomial system. The main contribution of the paper is a new algorithm to compute all isolated solutions of a system of polynomials. Standard techniques in computational algebra, often referred to as Stetter’s method [H.J. Stetter, Numerical Polynomial Algebra, SIAM, 2004], involve the computation of a Gröbner basis of the ideal generated by the polynomials and further require joint eigenvector computations in order to arrive at the zeros of the polynomial system. Our algorithm does neither require structural knowledge on the polynomial system, nor does it rely on the computation of joint eigenvectors.     

Convergence of the Lasserre hierarchy of SDP relaxations for convex polynomial programs without compactness

We show that the Lasserre hierarchy of semidefinite programming (SDP) relaxations with a slightly extended quadratic module for convex polynomial optimization problems always converges asymptotically even in the case of non-compact semi-algebraic feasible sets. We then prove that the positive definiteness of the Hessian of the associated Lagrangian at a saddle-point guarantees the finite convergence of the hierarchy. We do this by establishing a new sum-of-squares polynomial representation of convex polynomials over convex semi-algebraic sets.     

Semidefinite relaxations for semi-infinite polynomial programming

This paper studies how to solve semi-infinite polynomial programming (SIPP) problems by semidefinite relaxation methods. We first recall two SDP relaxation methods for solving polynomial optimization problems with finitely many constraints. Then we propose an exchange algorithm with SDP relaxations to solve SIPP problems with compact index set. At last, we extend the proposed method to SIPP problems with noncompact index set via homogenization. Numerical results show that the algorithm is efficient in practice.     

Enhancing RLT relaxations via a new class of semidefinite cuts

In this paper, we propose a mechanism to tighten Reformulation-Linearization Technique (RLT) based relaxations for solving nonconvex programming problems by importing concepts from semidefinite programming (SDP), leading to a new class of semidefinite cutting planes. Given an RLT relaxation, the usual nonnegativity restrictions on the matrix of RLT product variables is replaced by a suitable positive semidefinite constraint. Instead of relying on specific SDP solvers, the positive semidefinite stipulation is re-written to develop a semi-infinite linear programming representation of the problem, and an approach is developed that can be implemented using traditional optimization software. Specifically, the infinite set of constraints is relaxed, and members of this set are generated as needed via a separation routine in polynomial time. In essence, this process yields an RLT relaxation that is augmented with valid inequalities, which are themselves classes of RLT constraints that we call semidefinite cuts. These semidefinite cuts comprise a relaxation of the underlying semidefinite constraint. We illustrate this strategy by applying it to the case of optimizing a nonconvex quadratic objective function over a simplex. The algorithm has been implemented in C++, using CPLEX callable routines, and two types of semidefinite restrictions are explored along with several implementation strategies. Several of the most promising lower bounding strategies have been implemented within a branch-and-bound framework. Computational results indicate that the cutting plane algorithm provides a significant tightening of the lower bound obtained by using RLT alone. Moreover, when used within a branch-and-bound framework, the proposed lower bound significantly reduces the effort required to obtain globally optimal solutions.     

The equivalence of semidefinite relaxations of polynomial 0-1 and ± 1 programs via scaling

Many combinatorial optimization problems can be modelled as polynomial-programming problems in binary variables that are all 0-1 or ±1. A sufficient condition under which a common method for obtaining semidefinite-programming relaxations of the two models of the same problem gives equivalent relaxations is established.     

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.     

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.     

On the strength of recursive McCormick relaxations for binary polynomial optimization

Recursive McCormick relaxations are among the most popular convexification techniques for binary polynomial optimization. It is well-understood that both the quality and the size of these relaxations depend on the recursive sequence and finding an optimal sequence amounts to solving a difficult combinatorial optimization problem. We prove that any recursive McCormick relaxation is implied by the extended flower relaxation, a linear programming relaxation, which for binary polynomial optimization problems with fixed degree can be solved in strongly polynomial time.     

Linear polynomial constraints inferencing algorithm

This paper propose the inference mechanism for handling linear polynomial constraints called consistency checking algorithm based on the feasibility checking algorithm borrowed from linear programming. In contrast with other approaches, proposed algorithm can efficiently and coherented by linear polynomial forms. The developed algorithm is successfully applied to the symbolic simulation that offers a convenient means to conduct multiple, simultaneous exploration of model behaviors.     

Set packing relaxations of some integer programs

This paper is about set packing relaxations of combinatorial optimization problems associated with acyclic digraphs and linear orderings, cuts and multicuts, and set packings themselves. Families of inequalities that are valid for such a relaxation as well as the associated separation routines carry over to the problems under investigation. Received: September 1997 / Accepted: November 1999?Published online June 8, 2000     

Convex quadratic relaxations for mixed-integer nonlinear programs in power systems

This paper presents a set of new convex quadratic relaxations for nonlinear and mixed-integer nonlinear programs arising in power systems. The considered models are motivated by hybrid discrete/continuous applications where existing approximations do not provide optimality guarantees. The new relaxations offer computational efficiency along with minimal optimality gaps, providing an interesting alternative to state-of-the-art semidefinite programming relaxations. Three case studies in optimal power flow, optimal transmission switching and capacitor placement demonstrate the benefits of the new relaxations.     

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.     

Stabilizer-based symmetry breaking constraints for mathematical programs

Mathematical programs whose formulation is symmetric often take a long time to solve using Branch-and-Bound type algorithms, because of the several symmetric optima. A simple technique used in these cases is to adjoin symmetry breaking constraints to the formulation before solving the problem. These constraints: (a) aim to guarantee that at least one optimum is feasible, whilst making some of the symmetric optima infeasible; and (b) are usually associated to the different orbits of the action of the formulation group on the set of variable indices. In general, one cannot adjoin symmetry breaking constraints from more than one orbit. In Liberti (Math Program A 131:273–304, doi:10.1007/s10107-010-0351-0, 2012), some (restrictive) sufficient conditions are presented which make it possible to adjoin such constraints from several orbits at the same time. In this paper we present a new, less restrictive method for the same task, and show it performs better computationally.     

Duality for quasi-concave programs with explicit constraints

The idea of duality is now well established in the theory of concave programming. The basis of this duality is the concave conjugate transform. This has been exemplified in the development of generalised geometric programming. Much of the current research in duality theory is focused on relaxing the requirement of concavity. Here we develop a duality theory for mathematical programs with a quasi concave objective function and explicit quasi concave constraints. Generalisations of the concave conjugate transform are introduced which pair quasi concave functions as the concave conjugate transform does for concave functions. Optimality conditions are derived relating the primal quasi concave program to its dual. This duality theory was motivated by and has implications in certain problems of mathematical economics. An application to economics is given.     

Nonanticipative duality,relaxations, and formulations for chance-constrained stochastic programs

We propose two new Lagrangian dual problems for chance-constrained stochastic programs based on relaxing nonanticipativity constraints. We compare the strength of the proposed dual bounds and demonstrate that they are superior to the bound obtained from the continuous relaxation of a standard mixed-integer programming (MIP) formulation. For a given dual solution, the associated Lagrangian relaxation bounds can be calculated by solving a set of single scenario subproblems and then solving a single knapsack problem. We also derive two new primal MIP formulations and demonstrate that for chance-constrained linear programs, the continuous relaxations of these formulations yield bounds equal to the proposed dual bounds. We propose a new heuristic method and two new exact algorithms based on these duals and formulations. The first exact algorithm applies to chance-constrained binary programs, and uses either of the proposed dual bounds in concert with cuts that eliminate solutions found by the subproblems. The second exact method is a branch-and-cut algorithm for solving either of the primal formulations. Our computational results indicate that the proposed dual bounds and heuristic solutions can be obtained efficiently, and the gaps between the best dual bounds and the heuristic solutions are small.     

Faster,but weaker,relaxations for quadratically constrained quadratic programs

We introduce a new relaxation framework for nonconvex quadratically constrained quadratic programs (QCQPs). In contrast to existing relaxations based on semidefinite programming (SDP), our relaxations incorporate features of both SDP and second order cone programming (SOCP) and, as a result, solve more quickly than SDP. A downside is that the calculated bounds are weaker than those gotten by SDP. The framework allows one to choose a block-diagonal structure for the mixed SOCP-SDP, which in turn allows one to control the speed and bound quality. For a fixed block-diagonal structure, we also introduce a procedure to improve the bound quality without increasing computation time significantly. The effectiveness of our framework is illustrated on a large sample of QCQPs from various sources.