Deterministic approximation algorithms for sphere constrained homogeneous polynomial optimization problems

Due to their fundamental nature and numerous applications, sphere constrained polynomial optimization problems have received a lot of attention lately. In this paper, we consider three such problems: (i) maximizing a homogeneous polynomial over the sphere; (ii) maximizing a multilinear form over a Cartesian product of spheres; and (iii) maximizing a multiquadratic form over a Cartesian product of spheres. Since these problems are generally intractable, our focus is on designing polynomial-time approximation algorithms for them. By reducing the above problems to that of determining the L ₂-diameters of certain convex bodies, we show that they can all be approximated to within a factor of Ω((log n/n) ^d/2–1) deterministically, where n is the number of variables and d is the degree of the polynomial. This improves upon the currently best known approximation bound of Ω((1/n) ^d/2–1) in the literature. We believe that our approach will find further applications in the design of approximation algorithms for polynomial optimization problems with provable guarantees.     

An FPTAS for a single-item capacitated economic lot-sizing problem with monotone cost structure

We present a fully polynomial time approximation scheme (FPTAS) for a capacitated economic lot-sizing problem with a monotone cost structure. An FPTAS delivers a solution with a given relative error ɛ in time polynomial in the problem size and in 1/ɛ. Such a scheme was developed by van Hoesel and Wagelmans [8] for a capacitated economic lot-sizing problem with monotone concave (convex) production and backlogging cost functions. We omit concavity and convexity restrictions. Furthermore, we take advantage of a straightforward dynamic programming algorithm applied to a rounded problem.     

An analytic center quadratic cut method for the convex quadratic feasibility problem

 We consider a quadratic cut method based on analytic centers for two cases of convex quadratic feasibility problems. In the first case, the convex set is defined by a finite yet large number, N, of convex quadratic inequalities. We extend quadratic cut algorithm of Luo and Sun [3] for solving such problems by placing or translating the quadratic cuts directly through the current approximate center. We show that, in terms of total number of addition and translation of cuts, our algorithm has the same polynomial worst case complexity as theirs [3]. However, the total number of steps, where steps consist of (damped) Newton steps, function evaluations and arithmetic operations, required to update from one approximate center to another is , where ε is the radius of the largest ball contained in the feasible set. In the second case, the convex set is defined by an infinite number of certain strongly convex quadratic inequalities. We adapt the same quadratic cut method for the first case to the second one. We show that in this case the quadratic cut algorithm is a fully polynomial approximation scheme. Furthermore, we show that, at each iteration, k, the total number steps (as described above) required to update from one approximate center to another is at most , with ε as defined above. Received: April 2000 / Accepted: June 2002 Published online: September 5, 2002 Key words. convex quadratic feasibility problem – interior-point methods – analytic center – quadratic cuts – potential function     

Global optimization of mixed-integer bilevel programming problems

Two approaches that solve the mixed-integer nonlinear bilevel programming problem to global optimality are introduced. The first addresses problems mixed-integer nonlinear in outer variables and C²-nonlinear in inner variables. The second adresses problems with general mixed-integer nonlinear functions in outer level. Inner level functions may be mixed-integer nonlinear in outer variables, linear, polynomial, or multilinear in inner integer variables, and linear in inner continuous variables. This second approach is based on reformulating the mixed-integer inner problem as continuous via its vertex polyheral convex hull representation and solving the resulting nonlinear bilevel optimization problem by a novel deterministic global optimization framework. Computational studies illustrate proposed approaches.     

Note on the computational complexity ofj-radii of polytopes in ℝ n

We show that, for fixed dimensionn, the approximation of inner and outerj-radii of polytopes in ℝ ⁿ , endowed with the Euclidean norm, is in ℙ. Our method is based on the standard polynomial time algorithms for solving a system of polynomial inequalities over the reals in fixed dimension.     

Updating credal networks is approximable in polynomial time

Credal networks relax the precise probability requirement of Bayesian networks, enabling a richer representation of uncertainty in the form of closed convex sets of probability measures. The increase in expressiveness comes at the expense of higher computational costs. In this paper, we present a new variable elimination algorithm for exactly computing posterior inferences in extensively specified credal networks, which is empirically shown to outperform a state-of-the-art algorithm. The algorithm is then turned into a provably good approximation scheme, that is, a procedure that for any input is guaranteed to return a solution not worse than the optimum by a given factor. Remarkably, we show that when the networks have bounded treewidth and bounded number of states per variable the approximation algorithm runs in time polynomial in the input size and in the inverse of the error factor, thus being the first known fully polynomial-time approximation scheme for inference in credal networks.     

Parametric Integer Programming Algorithm for Bilevel Mixed Integer Programs

We consider discrete bilevel optimization problems where the follower solves an integer program with a fixed number of variables. Using recent results in parametric integer programming, we present polynomial time algorithms for pure and mixed integer bilevel problems. For the mixed integer case where the leader’s variables are continuous, our algorithm also detects whether the infimum cost fails to be attained, a difficulty that has been identified but not directly addressed in the literature. In this case, it yields a “better than fully polynomial time” approximation scheme with running time polynomial in the logarithm of the absolute precision. For the pure integer case where the leader’s variables are integer, and hence optimal solutions are guaranteed to exist, we present an algorithm which runs in polynomial time when the total number of variables is fixed.     

Description of 2-integer continuous knapsack polyhedra

In this paper we discuss the polyhedral structure of several mixed integer sets involving two integer variables. We show that the number of the corresponding facet-defining inequalities is polynomial on the size of the input data and their coefficients can also be computed in polynomial time using a known algorithm [D. Hirschberg, C. Wong, A polynomial-time algorithm for the knapsack problem with two variables, Journal of the Association for Computing Machinery 23 (1) (1976) 147–154] for the two integer knapsack problem. These mixed integer sets may arise as substructures of more complex mixed integer sets that model the feasible solutions of real application problems.     

First-order inertial algorithms involving dry friction damping

We present a Branch-and-Cut algorithm for a class of nonlinear chance-constrained mathematical optimization problems with a finite number of scenarios. Unsatisfied scenarios can enter a recovery mode. This class corresponds to problems that can be reformulated as deterministic convex mixed-integer nonlinear programming problems with indicator variables and continuous scenario variables, but the size of the reformulation is large and quickly becomes impractical as the number of scenarios grows. The Branch-and-Cut algorithm is based on an implicit Benders decomposition scheme, where we generate cutting planes as outer approximation cuts from the projection of the feasible region on suitable subspaces. The size of the master problem in our scheme is much smaller than the deterministic reformulation of the chance-constrained problem. We apply the Branch-and-Cut algorithm to the mid-term hydro scheduling problem, for which we propose a chance-constrained formulation. A computational study using data from ten hydroplants in Greece shows that the proposed methodology solves instances faster than applying a general-purpose solver for convex mixed-integer nonlinear programming problems to the deterministic reformulation, and scales much better with the number of scenarios.

     

Geometric Optimization Problems Likely Not Contained in $$\mathbb{A}\mathbb{P}\mathbb{X}$$

   Abstract. Maximizing geometric functionals such as the classical l _p -norms over polytopes plays an important role in many applications, hence it is desirable to know how efficiently the solutions can be computed or at least approximated. While the maximum of the l _∞ -norm over polytopes can be computed in polynomial time, for 2≤ p < ∞ the l _p -norm-maxima cannot be computed in polynomial time within a factor of 1.090 , unless P=NP. This result holds even if the polytopes are centrally symmetric parallelotopes. Quadratic Programming is a problem closely related to norm-maximization, in that in addition to a polytope P ⊂ R ⁿ , numbers c _ij , 1≤ i≤ j≤ n , are given and the goal is to maximize Σ _{1≤ i≤ j≤ n} c _ij x _i x _j over P . It is known that Quadratic Programming does not admit polynomial-time approximation within a constant factor, unless P=NP. Using the observation that the latter result remains true even if the existence of an integral optimal point is guaranteed, in this paper it is proved that analogous inapproximability results hold for computing the l _p -norm-maximum and various l _p -radii of centrally symmetric polytopes for 2≤ p < ∞.     

Geometric Optimization Problems Likely Not Contained in APX

Abstract. Maximizing geometric functionals such as the classical l _p -norms over polytopes plays an important role in many applications, hence it is desirable to know how efficiently the solutions can be computed or at least approximated. While the maximum of the l _∞ -norm over polytopes can be computed in polynomial time, for 2≤ p < ∞ the l _p -norm-maxima cannot be computed in polynomial time within a factor of 1.090 , unless P=NP. This result holds even if the polytopes are centrally symmetric parallelotopes. Quadratic Programming is a problem closely related to norm-maximization, in that in addition to a polytope P ⊂ R ⁿ , numbers c _ij , 1≤ i≤ j≤ n , are given and the goal is to maximize Σ _{1≤ i≤ j≤ n} c _ij x _i x _j over P . It is known that Quadratic Programming does not admit polynomial-time approximation within a constant factor, unless P=NP. Using the observation that the latter result remains true even if the existence of an integral optimal point is guaranteed, in this paper it is proved that analogous inapproximability results hold for computing the l _p -norm-maximum and various l _p -radii of centrally symmetric polytopes for 2≤ p < ∞.     

A polyhedral study of the cardinality constrained knapsack problem

 A cardinality constrained knapsack problem is a continuous knapsack problem in which no more than a specified number of nonnegative variables are allowed to be positive. This structure occurs, for example, in areas such as finance, location, and scheduling. Traditionally, cardinality constraints are modeled by introducing auxiliary 0-1 variables and additional constraints that relate the continuous and the 0-1 variables. We use an alternative approach, in which we keep in the model only the continuous variables, and we enforce the cardinality constraint through a specialized branching scheme and the use of strong inequalities valid for the convex hull of the feasible set in the space of the continuous variables. To derive the valid inequalities, we extend the concepts of cover and cover inequality, commonly used in 0-1 programming, to this class of problems, and we show how cover inequalities can be lifted to derive facet-defining inequalities. We present three families of non-trivial facet-defining inequalities that are lifted cover inequalities. Finally, we report computational results that demonstrate the effectiveness of lifted cover inequalities and the superiority of the approach of not introducing auxiliary 0-1 variables over the traditional MIP approach for this class of problems. Received: March 13, 2003 Published online: April 10, 2003 Key Words. mixed-integer programming – knapsack problem – cardinality constrained programming – branch-and-cut     

Solving certain singly constrained convex optimization problems in production planning

This paper considers the problem of minimizing a special convex function subject to one linear constraint. Based upon a theorem for lower and upper bounds on the Lagrange multiplier a fully polynomial time approximation scheme is proposed. The efficiency of the algorithm is demonstrated by a computational experiment.     

Hybrid approximation for minimum-cost target coverage in wireless sensor networks

This paper presents a hybrid approximation scheme for the Max-SNP-complete minimum-cost target coverage problem in wireless sensor networks. LP-rounding and set-cover selection are polynomial-time approximations for this problem. Our hybrid scheme combines these two methods using a crafted convex combination. We show that the hybrid scheme with appropriately chosen coefficients produces much better approximations than either of the two methods used alone. We show that, through a large number of numerical experiments, the hybrid scheme never exceeds an approximation ratio of 1.14, providing up to 14.86% improvement over the best approximations previously known.     

Error of Asymptotic Formulae for Volume Approximation of Convex Bodies in

 We estimate the error of asymptotic formulae for volume approximation of sufficiently differentiable convex bodies by circumscribed convex polytopes as the number of facets tends to infinity. Similar estimates hold for approximation with inscribed and general polytopes and for vertices instead of facets. Our result is then applied to estimate the minimum isoperimetric quotient of convex polytopes as the number of facets tends to infinity.     

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 more powerful ways of rounding. Moreover, 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. These linear complexity bounds give a substantial improvement of the best previously known polynomial bounds [A. Caprara, et al., Approximation algorithms for knapsack problems with cardinality constraints, European J. Oper. Res. 123 (2000) 333-345].     

An Outcome Space Branch and Bound-Outer Approximation Algorithm for Convex Multiplicative Programming

This article presents a new global solution algorithm for Convex Multiplicative Programming called the Outcome Space Algorithm. To solve a given convex multiplicative program (P _D), the algorithm solves instead an equivalent quasiconcave minimization problem in the outcome space of the original problem. To help accomplish this, the algorithm uses branching, bounding and outer approximation by polytopes, all in the outcome space of problem (P _D). The algorithm economizes the computations that it requires by working in the outcome space, by avoiding the need to compute new vertices in the outer approximation process, and, except for one convex program per iteration, by requiring for its execution only linear programming techniques and simple algebra.     

The Surface Area Deviation of the Euclidean Ball and a Polytope

While there is extensive literature on approximation of convex bodies by inscribed or circumscribed polytopes, much less is known in the case of generally positioned polytopes. Here we give upper and lower bounds for approximation of convex bodies by arbitrarily positioned polytopes with a fixed number of vertices or facets in the symmetric surface area deviation.     

Some fully polynomial time randomized approximation scheme based on an evolutionary algorithm

Considering the class of discrete optimization problems that satisfy the Woeginger conditions of existence of a fully polynomial-time approximation scheme, we propose a fully polynomial-time randomized approximation scheme based on an evolutionary algorithm.