Semidefinite Optimization Approach Research Articles

A Convex Optimization Approach for Computing Correlated Choice Probabilities With Many Alternatives

A popular discrete choice model that incorporates correlation information is the multinomial probit (MNP) model where the random utilities of the alternatives are chosen from a multivariate normal distribution. Computing the choice probabilities is challenging in the MNP model when the number of alternatives is large. Mishra et al. (IEEE Transactions on Automatic Control, 2012) have proposed a semidefinite optimization approach to compute choice probabilities for the distribution of the random utilities that maximizes expected agent utility given only the mean, variance, and covariance information. Their model is referred to as the cross moment (CMM) model. Computing the choice probabilities with many alternatives is challenging in the CMM model, since one needs to solve large-scale semidefinite programs. We develop a simpler formulation as a representative agent model by maximizing over the choice probabilities in the unit simplex where the objective function is the sum of the expected utilities and a strongly concave perturbation function. By characterizing the perturbation function for the CMM model and its gradient, we develop a simple first-order gradient method with inexact line search to compute choice probabilities. We establish local linear convergence of this algorithm under mild assumptions on the choice probabilities. An implication of our results is that inverting the choice probabilities to compute the mean utilities is straightforward given any positive-definite covariance matrix. Numerical experiments show that this method can compute choice probabilities for a large number of alternatives within a reasonable amount of time while explicitly capturing the correlation information. Comparisons with simulation methods for MNP and semidefinite programming methods for CMM indicate the efficacy of the method.

A Semidefinite Optimization Approach to Quadratic Fractional Optimization with a Strictly Convex Quadratic Constraint

In this paper we consider a fractional optimization problem that minimizes the ratio of two quadratic functions subject to a strictly convex quadratic constraint. First using the extension of Charnes-Cooper transformation, an equivalent homogenized quadratic reformulation of the problem is given. Then we show that under certain assumptions, it can be solved to global optimality using semidenite optimization relaxation in polynomial time.

A semidefinite optimization approach to the Target Visitation Problem

We propose an exact algorithm for the Target Visitation Problem (TVP). The (TVP) is a composition of the Linear Ordering Problem and the Traveling Salesman Problem. It has several military and non-military applications, where two important, often competing factors are the overall distance traveled (e.g. by an unmanned aerial vehicle) and the visiting sequence of the various targets or points of interest. Hence our algorithm can be used to find the optimal visiting sequence of various pre-determined targets. First we show that the (TVP) is a special Quadratic Position Problem. Building on this finding we propose an exact semidefinite optimization approach to tackle the (TVP) and finally demonstrate its efficiency on a variety of benchmark instances with up to 50 targets.

Semidefinite Optimization Providing Guaranteed Bounds on Linear Functionals of Solutions of Linear Integral Equations with Smooth Kernels

Based on recent progress on moment problems, semidefinite optimization approach is proposed for estimating upper and lower bounds on linear functionals defined on solutions of linear integral equations with smooth kernels. The approach is also suitable for linear integrodifferential equations with smooth kernels. Firstly, the primal problem with smooth kernel is converted to a series of approximative problems with Taylor polynomials obtained by expanding the smooth kernel. Secondly, two semidefinite programs (SDPs) are constructed for every approximative problem. Thirdly, upper and lower bounds on related functionals are gotten by applying SeDuMi 1.1R3 to solve the two SDPs. Finally, upper and lower bounds series obtained by solving two SDPs, respectively infinitely approach the exact value of discussed functional as approximative order of the smooth kernel increases. Numerical results show that the proposed approach is effective for the discussed problems.

Flow-Level Analysis of Energy Efficiency Performance for Device-to-Device Communications in OFDM Cellular Networks

In this paper, the energy efficiency performance of device-to-device (D2D) communications in orthogonal frequency division multiplexing cellular networks is studied. Different from previous work that was based on the static interference model, we assume a dynamic number of competing flows, such as continuous transfers of file transport protocol or web browsing sessions. The complex, dynamic interaction of the amount of backlogged traffic at the D2D and cellular links introduced by the strong impact of interference between them is captured by the coupled-processors server in the formulated model. The energy efficiency and spectral efficiency of the different resource-sharing strategies are analyzed using the semidefinite optimization approach. It is shown that the simulation results match well with the analytical bounds under different traffic loads and topologies.

A Semidefinite Optimization Approach to the Directed Circular Facility Layout Problem

We propose a new combinatorial optimization problem, the so-called Directed Circular Facility Layout Problem (DCFLP). The (DCFLP) seems to be quite interesting as it contains several other problems that have been discussed extensively in literature as special cases. We show that the (DCFLP) is closely related to the Single-Row Facility Layout Problem (SRFLP) and hence we adapt the leading algorithmic method for the (SRFLP) by suggesting an appropriate modelling approach for the (DCFLP). Finally we show that this algorithmic approach yields promising computational results on a variety of benchmark instances.

Output regulation of a class of continuous-time Markovian jumping systems

This paper studies output regulation for continuous-time Markovian jumping systems (MJSs), for which mode-dependent regulation equations are investigated. With the extension of regulation scheme to MJSs by stochastic Lyaponov–Krasovskii functional framework, sufficient conditions are, respectively, obtained for state feedback and error feedback. The resulting closed-loop system is guaranteed to be stochastically stable and the output regulation error almost asymptotically converges to zero. A semi-definite optimization approach via disciplined convex programming is adopted to ensure relaxed solutions of the regulation equations. Finally, two numerical simulations are given to illustrate the performance of the proposed approach.

Output tracking of Markovian jumping systems via error feedback regulation

AbstractThis paper studies the output tracking problem of continuous-time Markovian jumping systems (MJSs) via error feedback scheme. With the extension of output regulation to MJSs, sufficient conditions are obtained based on stochastic Lyapunov-Krasovskii functional. The resulting closed-loop system is guaranteed to be stochastically stable and the output tracking is achieved almost asymptotically. Moreover, the output regulation error almost asymptotically converges to zero. In order to ensure the relaxed solutions of the regulation equations, we described the problem as a semi-definite optimization approach via disciplined convex programming. Simulation result is also given to illustrate the performance and effectiveness of the proposed approach.

A semidefinite optimization approach to the steady-state analysis of queueing systems

Computing the steady-state distribution in Markov chains for general distributions and general state space is a computationally challenging problem. In this paper, we consider the steady-state stochastic model $\boldsymbol {W}\stackrel{d}{=}g(\boldsymbol {W},\boldsymbol {X})$ where the equality is in distribution. Given partial distributional information on the random variables X, we want to estimate information on the distribution of the steady-state vector W. Such models naturally occur in queueing systems, where the goal is to find bounds on moments of the waiting time under moment information on the service and interarrival times. In this paper, we propose an approach based on semidefinite optimization to find such bounds. We show that the classical Kingman’s and Daley’s bounds for the expected waiting time in a GI/GI/1 queue are special cases of the proposed approach. We also report computational results in the queueing context that indicate the method is promising.

A semidefinite optimization approach for the single-row layout problem with unequal dimensions

The facility layout problem is concerned with the arrangement of a given number of rectangular facilities so as to minimize the total cost associated with the (known or projected) interactions between them. We consider the one-dimensional space-allocation problem (ODSAP), also known as the single-row facility layout problem, which consists in finding an optimal linear placement of facilities with varying dimensions on a straight line. We construct a semidefinite programming (SDP) relaxation providing a lower bound on the optimal value of the ODSAP. To the best of our knowledge, this is the first non-trivial global lower bound for the ODSAP in the published literature. This SDP approach implicitly takes into account the natural symmetry of the problem and, unlike other algorithms in the literature, does not require the use of any explicit symmetry-breaking constraints. Furthermore, the structure of the SDP relaxation suggests a simple heuristic procedure which extracts a feasible solution to the ODSAP from the optimal matrix solution to the SDP relaxation. Computational results show that this heuristic yields a solution which is consistently within a few percentage points of the global optimal solution.

Optimal Inequalities in Probability Theory: A Convex Optimization Approach

We propose a semidefinite optimization approach to the problem of deriving tight moment inequalities for $P(X\in S)$, for a set S defined by polynomial inequalities and a random vector X defined on $\Omega\subseteq {\cal R}^n$ that has a given collection of up to kth-order moments. In the univariate case, we provide optimal bounds on $P(X\in S)$, when the first k moments of X are given, as the solution of a semidefinite optimization problem in k + 1 dimensions. In the multivariate case, if the sets S and $\Omega$ are given by polynomial inequalities, we obtain an improving sequence of bounds by solving semidefinite optimization problems of polynomial size in n, for fixed k. We characterize the complexity of the problem of deriving tight moment inequalities. We show that it is NP-hard to find tight bounds for $k\geq 4$ and $\Omega={\cal R}^n$ and for $k\geq 2$ and $\Omega={\cal R}^n_+,$ when the data in the problem is rational. For k = 1 and $\Omega={\cal R}^n_+$ we show that we can find tight upper bounds by solving n convex optimization problems when the set S is convex, and we provide a polynomial time algorithm when S and $\Omega$ are unions of convex sets, over which linear functions can be optimized efficiently. For the case k = 2 and $\Omega={\cal R}^n$, we present an efficient algorithm for finding tight bounds when S is a union of convex sets, over which convex quadratic functions can be optimized efficiently.