A Global Search Method for Discrete Stochastic Optimization

Solution quality of random search methods for discrete stochastic optimization

In this paper, we study the problem of mutual interference cancellation among secondary users (SUs) and interference control to primary users (PUs) in spectrum sharing underlay cognitive radio networks (CRNs). Multiple antennas are used at the secondary base station (SBS) to form multiple beams towards to individual SUs, and a set of SUs are selected to adapt to the beams. For the interference control to PUs, we study power allocation among SUs to guarantee the interference to PUs below a tolerable level while maximizing SUs' QoS. Based on these conditions, the problem of joint power allocation and beamforming with SUs selection is studied. Specifically, we emphasize on the condition of imperfect channel sensing. And we formulate the problem as a discrete stochastic optimization problem, then an efficient algorithm based on a discrete stochastic optimization method is proposed to solve the joint power allocation and beamforming with SUs selection problem. The proposed algorithm has fast convergence rate. Finally, simulation results are presented to demonstrate the performance of the proposed scheme.

Optimal allocation of simulation experiments in discrete stochastic optimization and approximative algorithms

Ultra-Wideband (UWB) communication is a promising technology for high data rate multimedia services in future wireless personal and home entertainment networks. To support heterogeneous multimedia applications with a wide variety of QoS requirements and maintain fairness among various traffic classes, a utility-based optimal scheduling problem for UWB networks is formulated. To fully explore the wide spectrum of UWB, its unique characteristics and features, such as the potential of allowing simultaneous transmissions and precision positioning, are considered for scheduling. The optimal schedul- ing problem of UWB network is significantly different from that in traditional networks. In particular, each link proceeds in a peer-to-peer manner, and the aggregate utilities achieved by con- current transmissions are random in nature. Thus, the optimal scheduling is formalized as a discrete stochastic optimization problem. We propose an exclusive-region based global searching algorithm (ER-GSA) to locate the global optimum efficiently. Extensive simulations demonstrate the effectiveness and efficiency of the ER-GSA.

Optimal stochastic single-machine-tardiness scheduling by stochastic branch-and-bound

Simulation-based optimization using simulated annealing with ranking and selection

We discuss the choice of the estimation of the optimal solution when random search methods are applied to solve discrete stochastic optimization problems. At the present time, such optimization methods usually estimate the optimal solution using either the feasible solution the method is currently exploring or the feasible solution visited most often so far by the method. We propose using all the observed objective function values generated as the random search method moves around the feasible region seeking an optimal solution to obtain increasingly more precise estimates of the objective function values at the different points in the feasible region. At any given time, the feasible solution that has the best estimated objective function value (largest one for maximization problems; the smallest one for minimization problems) is used as the estimate of the optimal solution. We discuss the advantages of using this approach for estimating the optimal solution and present numerical results showing that modifying an existing random search method to use tnhis approach for estimating the optimal soluation appears to yield improved performance. We also present sereval rate of convergence results for random search methods using our approach for estimating the optimal solution. One these random search methods is a new variant of the stochastic comparison method; in addition to specifying the rate of convergence of this method, we prove that it is guaranteed to converge almost surely to the set of global optimal solutions and present a result that demonstrates that this method is likely to perform well in practice.

Recent advances in interference cancellation and signal-processing techniques can enable full-duplex radios and multipacket reception (MPR), which will have significant impact on medium access control (MAC) schemes. In this paper, we propose a MAC scheme in unmanned aerial vehicle (UAV) ad-hoc networks with full-duplex radios and MPR. To efficiently handle the highly mobile environment of a UAV ad-hoc network, a token-based technique is used to update information in the network as well. The MAC schemes in the presence of perfect and imperfect channel state information (CSI) are formulated as a combinatorial optimization problem and a discrete stochastic optimization problem, respectively. Simulation results show the effectiveness of the proposed MAC.

We consider stochastic discrete optimization problems where the decision variables are nonnegative integers. We propose and analyze an online control scheme which transforms the problem into a surrogate continuous optimization problem and proceeds to solve the latter using standard gradient-based approaches, while simultaneously updating both the actual and surrogate system states. It is shown that the solution of the original problem is recovered as an element of the discrete state neighborhood of the optimal surrogate state. For the special case of separable cost functions, we show that this methodology becomes particularly efficient. Finally, convergence of the proposed algorithm is established under standard technical conditions; numerical results are included in the paper to illustrate the fast convergence of this approach.

A Survey of Lagrangean Techniques for Discrete Optimization

Consider a finite set of alternatives under risk which have multiple attributes. MARPI is an interactive computer-based procedure to find an efficient choice in the sense of linear expected utility. The choice is based on incomplete information about the decision maker's preferences which is elicited and processed in a sequential way. The information includes qualitative properties of the multivariate utility function such as monotonicity, risk aversion, and separability. Further, in case of an additively separable utility function, bounds on the scaling constants are elicited, and preferences (not necessarily indifferences) between sure amounts and lotteries are asked from the decision maker. The lotteries are Bernoulli lotteries generated by MARPI using special strategies. At every stage of the procedure the efficient set of alternatives is determined with respect to the information elicited so far.

We present a modification of the simulated annealing algorithm designed for solving discrete stochastic optimization problems. Like the original simulated annealing algorithm, our method has the hi...

A modification of the simulated annealing (SA) algorithm for solving discrete stochastic optimization problems where the objective function is stochastic and can be evaluated only through Monte Carlo simulation is proposed. In this modification, the Metropolis criterion depends on whether the objective function values indicate a statistically significant difference at each iteration. The differences between objective function values are considered to be statistically significant based on confidence intervals associated with these values. Unlike the original SA, the proposed method uses a constant temperature. It is shown that the configuration that has been visited most often in the first k iterations converges almost surely to a global optimizer. Computational results and comparisons with other SA algorithms are presented to demonstrate the performance of the proposed SA algorithm.

Population-based approaches have given us new search strategies and ideas in order to solve optimization problems. Usually, these methods are based on the performance carried out by a finite number of agents, which by the interaction between them they evolve and work all over the search space. Also, it is well-known that the correct employment of parameter values in this kind of method can positively impact their performance and behavior. In this context, the present work focuses on the design of a hybrid architecture which smartly balances the population size on run-time. In order to smartly balance and control the population size, a modular approach, named Linear Modular Population Balancer (LMPB), is proposed. The main ideas behind the designed architecture include the solving strategy behind a population-based metaheuristic, the influence of learning components based on multiple statistical modeling methods which transform the dynamic data generated into knowledge, and the possibilities to tackle both discrete and continuous optimization problems. In this regard, three modules are proposed for LMPB, which concern tasks such as the management of the population-based algorithm, parameter setting, probabilities, learning methods, and selection mechanism for the population size to employ. In order to test the viability and effectiveness of our proposed approach, we solve a set of well-known benchmark functions and the multidimensional knapsack problem (MKP). Additionally, we illustrate promising solving results, compare them against state-of-the-art methods which have proved to be good options for solving optimization problems, and give solid arguments for future work in the necessity to keep evolving this type of proposed architecture.

CHAPTER 2.8 - Convexity and Discrete Optimization