Integer Programming Subject to Monomial Constraints

We investigate integer programs containing monomial constraints. Due to the number-theoretic nature of these constraints, standard methods based on linear algebra cannot be applied directly. Instead, we present a reformulation resulting in integer programs with linear constraints and polynomial objective functions, using prime decompositions of the right hand sides. Moreover, we show that minimizing a linear objective function with nonnegative coefficients over bivariate constraints is possible in polynomial time

Solving unconstrained 0-1 polynomial programs through quadratic convex reformulation

We propose a solution approach for the problem (P) of minimizing an unconstrained binary polynomial optimization problem. We call this method PQCR (Polynomial Quadratic Convex Reformulation). The resolution is based on a 3-phase method. The first phase consists in reformulating (P) into a quadratic program (QP). For this, we recursively reduce the degree of (P) to two, by use of the standard substitution of the product of two variables by a new one. We then obtain a linearly constrained binary program. In the second phase, we rewrite the quadratic objective function into an equivalent and parametrized quadratic function using the equality x 2 i = x i and new valid quadratic equalities. Then, we focus on finding the best parameters to get a quadratic convex program which continuous relaxation's optimal value is maximized. For this, we build a semidefinite relaxation (SDP) of (QP). Then, we prove that the standard linearization inequalities, used for the quadratization step, are redundant in (SDP) in presence of the new quadratic equalities. Next, we deduce our optimal parameters from the dual optimal solution of (SDP). The third phase consists in solving (QP *), the optimal reformulated problem, with a standard solver. In particular, at each node of the branch-and-bound, the solver computes the optimal value of a continuous quadratic convex program. We present computational results on instances of the image restoration problem and of the low autocorrelation binary sequence problem. We compare PQCR with other convexification methods, and with the general solver Baron 17.4.1 [39]. We observe that most of the considered instances can be solved with our approach combined with the use of Cplex [24]

On the Feasibility of 5G Slice Resource Allocation With Spectral Efficiency: A Probabilistic Characterization

An important concern that 5G networks face is supporting a wide range of services and use cases with heterogeneous requirements. Radio access network (RAN) slices, understood as isolated virtual networks that share a common infrastructure, are a possible answer to this very demanding scenario and enable virtual operators to provide differentiated services over independent logical entities. This article addresses the feasibility of forming 5G slices, answering the question of whether the available capacity (resources) is sufficient to satisfy slice requirements. As spectral efficiency is one of the key metrics in 5G networks, we introduce the minislot-based slicing allocation (MISA) model, a novel 5G slice resource allocation approach that combines the utilization of both complete slots (or physical resource blocks) and mini-slots with the adequate physical layer design and service requirement constraints. We advocate for a probabilistic characterization that allows to estimate feasibility and characterize the behavior of the constraints, while an exhaustive search is very computationally demanding and the methods to check feasibility provide no information on the constraints. In such a characterization, the concept of phase transition allows for the identification of a clear frontier between the feasible and infeasible regions. Our method relies on an adaptation of the Wang-Landau algorithm to determine the existence of, at least, one solution to the problem. The conducted simulations show a significant improvement in spectral efficiency and feasibility of the MISA approach compared to the slot-based formulation, the identification of the phase transition, and valuable results to characterize the satisfiability of the constraints.The work of J. J. Escudero-Garzás was supported in part by the Spanish National Project TERESA-ADA (MINECO/AEI/FEDER, UE) under Grant TEC2017-90093-C3-2-R, and in part by the National Spectrum Consortium, USA, under Project NSC-16-0140

Nonlinear Integer Programming

Research efforts of the past fifty years have led to a development of linear integer programming as a mature discipline of mathematical optimization. Such a level of maturity has not been reached when one considers nonlinear systems subject to integrality requirements for the variables. This chapter is dedicated to this topic. The primary goal is a study of a simple version of general nonlinear integer problems, where all constraints are still linear. Our focus is on the computational complexity of the problem, which varies significantly with the type of nonlinear objective function in combination with the underlying combinatorial structure. Numerous boundary cases of complexity emerge, which sometimes surprisingly lead even to polynomial time algorithms. We also cover recent successful approaches for more general classes of problems. Though no positive theoretical efficiency results are available, nor are they likely to ever be available, these seem to be the currently most successful and interesting approaches for solving practical problems. It is our belief that the study of algorithms motivated by theoretical considerations and those motivated by our desire to solve practical instances should and do inform one another. So it is with this viewpoint that we present the subject, and it is in this direction that we hope to spark further research.Comment: 57 pages. To appear in: M. J\"unger, T. Liebling, D. Naddef, G. Nemhauser, W. Pulleyblank, G. Reinelt, G. Rinaldi, and L. Wolsey (eds.), 50 Years of Integer Programming 1958--2008: The Early Years and State-of-the-Art Surveys, Springer-Verlag, 2009, ISBN 354068274

Efficient Reduction of Polynomial Zero-One Optimization to the Quadratic Case

We address the problem of optimizing a polynomial with real coefficients over binary variables. We show that a complete polyhedral description of the linearization of such a problem can be derived in a simple way from the polyhedral description of the linearization of some quadratic optimization problem. The number of variables in the latter linearization is only slightly larger than in the former. If polynomial constraints are present in the original problem, then their linearized counterparts carry over to the linearized quadratic problem unchanged. If the original problem formulation doesn't contain any constraints, we obtain a reduction to unconstrained quadratic zero-one optimization, which is equivalent to the well-studied max-cut problem. The separation problem for general unconstrained polynomial zero-one optimization thus reduces to the separation problem for the cut polytope. This allows to transfer the entire knowledge gained for the latter polytope by intensive research and, in particular, the sophisticated separation techniques that have been developed. We report preliminary experimental results obtained with a straightforward implementation of this approach