Integer Polynomial Optimization in Fixed Dimension

We classify, according to their computational complexity, integer optimization problems whose constraints and objective functions are polynomials with integer coefficients and the number of variables is fixed. For the optimization of an integer polynomial over the lattice points of a convex polytope, we show an algorithm to compute lower and upper bounds for the optimal value. For polynomials that are non-negative over the polytope, these sequences of bounds lead to a fully polynomial-time approximation scheme for the optimization problem.Comment: In this revised version we include a stronger complexity bound on our algorithm. Our algorithm is in fact an FPTAS (fully polynomial-time approximation scheme) to maximize a non-negative integer polynomial over the lattice points of a polytop

Scheduling under Linear Constraints

We introduce a parallel machine scheduling problem in which the processing times of jobs are not given in advance but are determined by a system of linear constraints. The objective is to minimize the makespan, i.e., the maximum job completion time among all feasible choices. This novel problem is motivated by various real-world application scenarios. We discuss the computational complexity and algorithms for various settings of this problem. In particular, we show that if there is only one machine with an arbitrary number of linear constraints, or there is an arbitrary number of machines with no more than two linear constraints, or both the number of machines and the number of linear constraints are fixed constants, then the problem is polynomial-time solvable via solving a series of linear programming problems. If both the number of machines and the number of constraints are inputs of the problem instance, then the problem is NP-Hard. We further propose several approximation algorithms for the latter case.Comment: 21 page

On the complexity of nonlinear mixed-integer optimization

This is a survey on the computational complexity of nonlinear mixed-integer optimization. It highlights a selection of important topics, ranging from incomputability results that arise from number theory and logic, to recently obtained fully polynomial time approximation schemes in fixed dimension, and to strongly polynomial-time algorithms for special cases.Comment: 26 pages, 5 figures; to appear in: Mixed-Integer Nonlinear Optimization, IMA Volumes, Springer-Verla

Joint Scheduling and Resource Allocation in the OFDMA Downlink: Utility Maximization under Imperfect Channel-State Information

We consider the problem of simultaneous user-scheduling, power-allocation, and rate-selection in an OFDMA downlink, with the goal of maximizing expected sum-utility under a sum-power constraint. In doing so, we consider a family of generic goodput-based utilities that facilitate, e.g., throughput-based pricing, quality-of-service enforcement, and/or the treatment of practical modulation-and-coding schemes (MCS). Since perfect knowledge of channel state information (CSI) may be difficult to maintain at the base-station, especially when the number of users and/or subchannels is large, we consider scheduling and resource allocation under imperfect CSI, where the channel state is described by a generic probability distribution. First, we consider the "continuous" case where multiple users and/or code rates can time-share a single OFDMA subchannel and time slot. This yields a non-convex optimization problem that we convert into a convex optimization problem and solve exactly using a dual optimization approach. Second, we consider the "discrete" case where only a single user and code rate is allowed per OFDMA subchannel per time slot. For the mixed-integer optimization problem that arises, we discuss the connections it has with the continuous case and show that it can solved exactly in some situations. For the other situations, we present a bound on the optimality gap. For both cases, we provide algorithmic implementations of the obtained solution. Finally, we study, numerically, the performance of the proposed algorithms under various degrees of CSI uncertainty, utilities, and OFDMA system configurations. In addition, we demonstrate advantages relative to existing state-of-the-art algorithms

Analysis and implementation of fractional-order chaotic system with standard components

This paper is devoted to the problem of uncertainty in fractional-order Chaotic systems implemented by means of standard electronic components. The fractional order element (FOE) is typically substituted by one complex impedance network containing a huge number of discrete resistors and capacitors. In order to balance the complexity and accuracy of the circuit, a sparse optimization based parameter selection method is proposed. The random error and the uncertainty of system implementation are analyzed through numerical simulations. The effectiveness of the method is verified by numerical and circuit simulations, tested experimentally with electronic circuit implementations. The simulations and experiments show that the proposed method reduces the order of circuit systems and finds a minimum number for the combination of commercially available standard components.This work was supported in part by the National Natural Science Foundation of China under Grant 61501385, in part by the National Nuclear Energy Development Project of State Administration for Science, Technology and Industry for National Defense, PRC under Grant 18zg6103, and in part by Sichuan Science and Technology Program under Grant 2018JY0522. We would like to thank Xinghua Feng for meaningful discussion.info:eu-repo/semantics/publishedVersio