Polyhedral Relaxations Research Articles

Constrained global optimization for wine blending

Assemblage consists in blending base wines in order to create target wines. Recent developments in aroma analysis allow us to measure chemical compounds impacting the taste of wines. This chemical analysis makes it possible to design a decision tool for the following problem: given a set of target wines, determine which volumes must be extracted from each base wine to produce wines that satisfy constraints on aroma concentration, volumes, alcohol contents and price. This paper describes the modeling of wine assemblage as a mixed constrained optimization problem, where the main goal is to minimize the gap to the desired concentrations for every aromatic criterion. The deterministic branch and bound solvers Couenne and IbexOpt behave well on the wine blending problem thanks to their interval constraint propagation/programming and polyhedral relaxation methods. We also study the performance of other optimization goals that could be embedded in a configuration tool, where the different possible interactions amount to solving the same constraints with different objective functions. We finally show on a recent generic wine blending instance that the proposed optimization process scales up well with the number of base wines.

Global optimization with spline constraints: a new branch-and-bound method based on B-splines

This paper discusses the use of splines as constraints in mathematical programming. By combining the mature theory of the B-spline and the widely used branch-and-bound framework a novel spatial branch-and-bound (sBB) method is obtained. The method solves nonconvex mixed-integer nonlinear programming (MINLP) problems with spline constraints to global optimality. A broad applicability follows from the fact that a spline may represent any (piecewise) polynomial and accurately approximate other nonlinear functions. The method relies on a reformulation---convexification technique which results in lifted polyhedral relaxations that are efficiently solved by an LP solver. The method has been implemented in the sBB solver Convex ENvelopes for Spline Optimization (CENSO). In this paper CENSO is compared to several state-of-the-art MINLP solvers on a set of polynomially constrained NLP problems. To further display the versatility of the method a realistic pump synthesis problem of class MINLP is solved with exact and approximated pump characteristics.

Computational results of a semidefinite branch-and-bound algorithm for k-cluster

This computational paper presents a method to solve k-cluster problems exactly by intersecting semidefinite and polyhedral relaxations. Our algorithm uses a generic branch-and-bound method featuring an improved semidefinite bounding procedure. Extensive numerical experiments show that this algorithm outperforms the best known methods both in time and ability to solve large instances. For the first time, numerical results are reported for k-cluster problems on unstructured graphs with 160 vertices.

On the Optimality of Treating Interference as Noise

It is shown that in the K-user interference channel, if for each user the desired signal strength is no less than the sum of the strengths of the strongest interference from this user and the strongest interference to this user (all values in decibel scale), then the simple scheme of using point-to-point Gaussian codebooks with appropriate power levels at each transmitter and treating interference as noise (TIN) at every receiver (in short, TIN scheme) achieves all points in the capacity region to within a constant gap. The generalized degrees of freedom (GDoF) region under this condition is a polyhedron, which is shown to be fully achieved by the same scheme, without the need for time-sharing. The results are proved by first deriving a polyhedral relaxation of the GDoF region achieved by TIN, and then providing a dual characterization of this polyhedral region via the use of potential functions, and finally proving the optimality of this region in the desired regime.

On the polytope faces of the graph approximation problem

We study the polytope of the graph approximation problem. Some polyhedral relaxation of this polytope is built. We describe the class of valid inequalities for this polytope among which the inequalities that generate the facets are allocated.

A Linear-Programming Approximation of AC Power Flows

Linear active-power-only power flow approximations are pervasive in the planning and control of power systems. However, AC power systems are governed by a system of nonlinear nonconvex power flow equations. Existing linear approximations fail to capture key power flow variables, including reactive power and voltage magnitudes, both of which are necessary in many applications that require voltage management and AC power flow feasibility. This paper proposes novel linear-programming models (the LPAC models) that incorporate reactive power and voltage magnitudes in a linear power flow approximation. The LPAC models are built on a polyhedral relaxation of the cosine terms in the AC equations as well as Taylor approximations of the remaining nonlinear terms. Experimental comparisons with AC solutions on a variety of standard IEEE and Matpower benchmarks show that the LPAC models produce accurate values for active and reactive power, phase angles, and voltage magnitudes. The potential benefits of the LPAC models are illustrated on two “proof-of-concept” studies in power restoration and capacitor placement.

Semidefinite relaxations of ordering problems

Ordering problems assign weights to each ordering and ask to find an ordering of maximum weight. We consider problems where the cost function is either linear or quadratic. In the first case, there is a given profit if the element $$u$$ is before $$v$$ in the ordering. In the second case, the profit depends on whether $$u$$ is before $$v$$ and $$r$$ is before $$s$$ . The linear ordering problem is well studied, with exact solution methods based on polyhedral relaxations. The quadratic ordering problem does not seem to have attracted similar attention. We present a systematic investigation of semidefinite optimization based relaxations for the quadratic ordering problem, extending and improving existing approaches. We show the efficiency of our relaxations by providing computational experience on a variety of problem classes.

Optimization-based Domain Reduction in Guaranteed Parameter Estimation of Nonlinear Dynamic Systems

This paper is concerned with guaranteed parameter estimation in nonlinear dynamic systems in a context of bounded measurement error. The problem consists of finding—or approximating as closely as possible—the set of all possible parameter values such that the predicted outputs match the corresponding measurements within prescribed error bounds. An exhaustive search procedure is applied, whereby the parameter set is successively partitioned into smaller boxes and exclusion tests are performed to eliminate some of these boxes, until a prespecified threshold on the approximation level is met. In order to enhance the convergence of this procedure, we investigate the use of optimization-based domain reduction techniques for tightening the parameter boxes before partitioning. We construct such bound-reduction problems as linear programs from the polyhedral relaxation of Taylor models of the predicted outputs. When applied to a simple case study, the proposed approach is found to reduce the computational burden significantly, both in terms of CPU time and number of iterations.

Valid Inequalities and Convex Hulls for Multilinear Functions

We study the convex hull of the bounded, nonconvex set of a product of n variables for any n ≥ 2. We seek to derive strong valid linear inequalities for this set, which we call M_n; this is motivated by the fact that many exact solvers for nonconvex problems use polyhedral relaxations so as to compute a lower bound via linear programming solvers. We present a class of linear inequalities that, together with the well-known McCormick inequalities, defines the convex hull of M_2. This class of inequalities, which we call lifted tangent inequalities, is uncountably infinite, which is not surprising given that the convex hull of M_n is not a polyhedron. This class of inequalities generalizes directly to M_n for n > 2, allowing us to define strengthened relaxations for these higher dimensional sets as well.

A partitioning algorithm for the network loading problem

This paper proposes a Benders-like partitioning algorithm to solve the network loading problem. The approach is an iterative method in which the integer programming solver is not used to produce the best integer point in the polyhedral relaxation of the set of feasible capacities. Rather, it selects an integer solution that is closest to the best known integer solution. Contrary to previous approaches, the method does not exploit the original mixed integer programming formulation of the problem. The effort of computing integer solutions is entirely left to a pure integer programming solver while valid inequalities are generated by solving standard nonlinear multicommodity flow problems. The method is compared to alternative approaches proposed in the literature and appears to be efficient for computing good upper bounds.

Multiterm polyhedral relaxations for nonconvex, quadratically constrained quadratic programs

This article addresses the generation of strong polyhedral relaxations for nonconvex, quadratically constrained quadratic programs (QCQPs). Using the convex envelope of multilinear functions as our starting point, we develop a polyhedral relaxation for QCQP, along with a cutting plane algorithm for its implementation. Our relaxations are multiterm, i.e. they are derived from the convex envelope of the sum of multiple bilinear terms of quadratic constraints, thereby providing tighter bounds than the standard termwise relaxation of the bilinear functions. Computational results demonstrate the usefulness of the proposed cutting planes.

A Lifted Linear Programming Branch-and-Bound Algorithm for Mixed-Integer Conic Quadratic Programs

This paper develops a linear-programming-based branch-and-bound algorithm for mixed-integer conic quadratic programs. The algorithm is based on a known higher-dimensional or lifted polyhedral relaxation of conic quadratic constraints. The algorithm is different from other linear-programming-based branch-and-bound algorithms for mixed-integer nonlinear programs in that it is not based on cuts from gradient inequalities and it sometimes branches on integer feasible solutions. The algorithm is tested on a series of portfolio optimization problems. It is shown that it significantly outperforms commercial and open-source solvers based on both linear and nonlinear relaxations.

Solving Max-Cut to optimality by intersecting semidefinite and polyhedral relaxations

We present a method for finding exact solutions of Max-Cut, the problem of finding a cut of maximum weight in a weighted graph. We use a Branch-and-Bound setting that applies a dynamic version of the bundle method as bounding procedure. This approach uses Lagrangian duality to obtain a “nearly optimal” solution of the basic semidefinite Max-Cut relaxation, strengthened by triangle inequalities. The expensive part of our bounding procedure is solving the basic semidefinite relaxation of the Max-Cut problem, which has to be done several times during the bounding process. We review other solution approaches and compare the numerical results with our method. We also extend our experiments to instances of unconstrained quadratic 0–1 optimization and to instances of the graph equipartition problem. The experiments show that our method nearly always outperforms all other approaches. In particular, for dense graphs, where linear programming-based methods fail, our method performs very well. Exact solutions are obtained in a reasonable time for any instance of size up to n = 100, independent of the density. For some problems of special structure we can solve even larger problem classes. We could prove optimality for several problems of the literature where, to the best of our knowledge, no other method is able to do so.

Lattice Based Extended Formulations for Integer Linear Equality Systems

We study different extended formulations for the set X = {x E Zn | Ax = Ax0} in order to tackle the feasibility problem for the set X+ = X N Zn+. Here the goal is not to find an improved polyhedral relaxation of conv(X+), but rather to reformulate in such a way that the new variables introduced provide good branching directions, and in certain circumstances permit one to deduce rapidly that the instance is infeasible. For the case that A has one row a we analyze the reformulations in more detail. In particular, we determine the integer width of the extended formulations in the direction of the last coordinate, and derive a lower bound on the Frobenius number of a. We also suggest how a decomposition of the vector a can be obtained that will provide a useful extended formulation. Our theoretical results are accompanied by a small computational study.

Effective Relaxations and Partitioning Schemes for Solving Water Distribution Network Design Problems to Global Optimality

In this paper, we address the development of a global optimization procedure for the problem of designing a water distribution network, including the case of expanding an already existing system, that satisfies specified flow demands at stated pressure head requirements. The proposed approach significantly improves upon a previous method of Sherali et al. (1998) by way of adopting tighter polyhedral relaxations, and more effective partitioning strategies in concert with a maximal spanning tree-based branching variable selection procedure. Computational experience on three standard test problems from the literature is provided to evaluate the proposed procedure. For all these problems, proven global optimal solutions within a tolerance of 10−4% and/or within 1$ of optimality are obtained. In particular, the two larger instances of the Hanoi and the New York test networks are solved to global optimality for the very first time in the literature. A new real network design test problem based on the Town of Blacksburg Water Distribution System is also offered to be included in the available library of test cases, and related computational results are presented.

Solving Graph Bisection Problems with Semidefinite Programming

An exact solution method for the graph bisection problem is presented. We describe a branch-and-bound algorithm which is based on a cutting plane approach combining semidefinite programming and polyhedral relaxations. We report on extensive numerical experiments which were performed for various classes of graphs. The results indicate that the present approach solves general problem instances with 80–90 vertices exactly in reasonable time and provides tight approximations for larger instances. Our approach is particularly well suited for special classes of graphs as planar graphs and graphs based on grid structures.

On the Convergence of Sequences of Convex Sets in Finite Dimensions

Four types of convergence for sequences of convex sets are investigated. Their interrelationships are explored.