A dynamic inequality generation scheme for polynomial programming

Hierarchies of semidefinite programs have been used to approximate or even solve polynomial programs. This approach rapidly becomes computationally expensive and is often tractable only for problems of small size. In this paper, we propose a dynamic inequality generation scheme to generate valid polynomial inequalities for general polynomial programs. When used iteratively, this scheme improves the bounds without incurring an exponential growth in the size of the relaxation. As a result, the proposed scheme is in principle scalable to large general polynomial programming problems. When all the variables of the problem are non-negative or when all the variables are binary, the general algorithm is specialized to a more efficient algorithm. In the case of binary polynomial programs, we show special cases for which the proposed scheme converges to the global optimal solution. We also present several examples illustrating the computational behavior of the scheme and provide comparisons with Lasserre’s approach and, for the binary linear case, with the lift-and-project method of Balas, Ceria, and Cornuejols

New Conic Optimization Techniques for Solving Binary Polynomial Programming Problems

Polynomial programming, a class of non-linear programming where the objective and the constraints are multivariate polynomials, has attracted the attention of many researchers in the past decade. Polynomial programming is a powerful modeling tool that captures various optimization models. Due to the wide range of applications, a research topic of high interest is the development of computationally efficient algorithms for solving polynomial programs. Even though some solution methodologies are already available and have been studied in the literature, these approaches are often either problem specific or are inapplicable for large-scale polynomial programs. Most of the available methods are based on using hierarchies of convex relaxations to solve polynomial programs; these schemes grow exponentially in size becoming rapidly computationally expensive. The present work proposes methods and implementations that are capable of solving polynomial programs of large sizes. First we propose a general framework to construct conic relaxations for binary polynomial programs, this framework allows us to re-derive previous relaxation schemes and provide new ones. In particular, three new relaxations for binary quadratic polynomial programs are presented. The first two relaxations, based on second-order cone and semidefinite programming, represent a significant improvement over previous practical relaxations for several classes of non-convex binary quadratic polynomial problems. The third relaxation is based purely on second-order cone programming, it outperforms the semidefinite-based relaxations that are proposed in the literature in terms of computational efficiency while being comparable in terms of bounds. To strengthen the relaxations further, a dynamic inequality generation scheme to generate valid polynomial inequalities for general polynomial programs is presented. When used iteratively, this scheme improves the bounds without incurring an exponential growth in the size of the relaxation. The scheme can be used on any initial relaxation of the polynomial program whether it is second-order cone based or semidefinite based relaxations. The proposed scheme is specialized for binary polynomial programs and is in principle scalable to large general combinatorial optimization problems. In the case of binary polynomial programs, the proposed scheme converges to the global optimal solution under mild assumptions on the initial approximation of the binary polynomial program. Finally, for binary polynomial programs the proposed relaxations are integrated with the dynamic scheme in a branch-and-bound algorithm to find global optimal solutions

Hybrid Rounding Techniques for Knapsack Problems

We address the classical knapsack problem and a variant in which an upper bound is imposed on the number of items that can be selected. We show that appropriate combinations of rounding techniques yield novel and powerful ways of rounding. As an application of these techniques, we present a linear-storage Polynomial Time Approximation Scheme (PTAS) and a Fully Polynomial Time Approximation Scheme (FPTAS) that compute an approximate solution, of any fixed accuracy, in linear time. This linear complexity bound gives a substantial improvement of the best previously known polynomial bounds.Comment: 19 LaTeX page

Reformulation and decomposition of integer programs

In this survey we examine ways to reformulate integer and mixed integer programs. Typically, but not exclusively, one reformulates so as to obtain stronger linear programming relaxations, and hence better bounds for use in a branch-and-bound based algorithm. First we cover in detail reformulations based on decomposition, such as Lagrangean relaxation, Dantzig-Wolfe column generation and the resulting branch-and-price algorithms. This is followed by an examination of Benders’ type algorithms based on projection. Finally we discuss in detail extended formulations involving additional variables that are based on problem structure. These can often be used to provide strengthened a priori formulations. Reformulations obtained by adding cutting planes in the original variables are not treated here.Integer program, Lagrangean relaxation, column generation, branch-and-price, extended formulation, Benders' algorithm

Counting approximately-shortest paths in directed acyclic graphs

Given a directed acyclic graph with positive edge-weights, two vertices s and t, and a threshold-weight L, we present a fully-polynomial time approximation-scheme for the problem of counting the s-t paths of length at most L. We extend the algorithm for the case of two (or more) instances of the same problem. That is, given two graphs that have the same vertices and edges and differ only in edge-weights, and given two threshold-weights L_1 and L_2, we show how to approximately count the s-t paths that have length at most L_1 in the first graph and length at most L_2 in the second graph. We believe that our algorithms should find application in counting approximate solutions of related optimization problems, where finding an (optimum) solution can be reduced to the computation of a shortest path in a purpose-built auxiliary graph

Optimal Data Placement on Networks With Constant Number of Clients

We introduce optimal algorithms for the problems of data placement (DP) and page placement (PP) in networks with a constant number of clients each of which has limited storage availability and issues requests for data objects. The objective for both problems is to efficiently utilize each client's storage (deciding where to place replicas of objects) so that the total incurred access and installation cost over all clients is minimized. In the PP problem an extra constraint on the maximum number of clients served by a single client must be satisfied. Our algorithms solve both problems optimally when all objects have uniform lengths. When objects lengths are non-uniform we also find the optimal solution, albeit a small, asymptotically tight violation of each client's storage size by $\epsilon$lmax where lmax is the maximum length of the objects and $\epsilon$ some arbitrarily small positive constant. We make no assumption on the underlying topology of the network (metric, ultrametric etc.), thus obtaining the first non-trivial results for non-metric data placement problems