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This paper discusses the computation of matrix chain products of the form M1 X M2 X ... X Mn where Mi's are matrices. The order in which the matrices are computed affects the number of operations. A sufficient condition about the association of the matrices in the optimal order is presented. An O( n) algorithm to find an order of computation which takes less than 25 percent longer than the optimal time Topt is also presented. In most cases, the algorithm yields the optimal order or an order which takes only a few percent longer than Topt (less than I percent on the average). [ABSTRACT FROM AUTHOR]

This article presents abstracts of mathematical software. It includes "Multiprecision Integer Division Examples Using Arbitrary Radix," "Procedures for Optimization Problems with a Mixture of Bounds and General Linear Constraints," and "A Program Complex for Solving Systems of Linear Algebraic Equations."

In this paper, a time-varying distributed convex optimization problem is studied for continuous-time multi-agent systems. The objective is to minimize the sum of local time-varying cost functions, each of which is known to only an individual agent, through local interaction. Here, the optimal point is time varying and creates an optimal trajectory. Control algorithms are designed for the cases of single-integrator and double-integrator dynamics. In both cases, a centralized approach is first introduced to solve the optimization problem. Then, this problem is solved in a distributed manner and a discontinuous algorithm based on the signum function is proposed in each case. In the case of single-integrator (respectively, double-integrator) dynamics, each agent relies only on its own position and the relative positions (respectively, positions and velocities) between itself and its neighbors. A gain adaption scheme is introduced in both algorithms to eliminate certain global information requirement. To relax the restricted assumption imposed on feasible cost functions, an estimator based algorithm using the signum function is proposed, where each agent uses dynamic average tracking as a tool to estimate the centralized control input. As a tradeoff, the estimator-based algorithm necessitates communication between neighbors. Then, in the case of double-integrator dynamics, the proposed algorithms are further extended. Two continuous algorithms based on, respectively, a time-varying and a fixed boundary layer are proposed as continuous approximations of the signum function. To account for interagent collision for physical agents, a distributed convex optimization problem with swarm tracking behavior is introduced for both single-integrator and double-integrator dynamics. It is shown that the center of the agents tracks the optimal trajectory, the connectivity of the agents is maintained, and interagent collision is avoided. [ABSTRACT FROM AUTHOR]

This paper considers the infinite horizon optimal control of logical control networks, including Boolean control networks as a special case. Using the framework of game theory, the optimal control problem is formulated. In the sight of the algebraic form of a logical control network, its cycles can be calculated algebraically. Then the optimal control is revealed over a certain cycle. When the games, using memory \mu>1 (which means the players only consider previous \mu steps' action at each step), are considered, the higher order logical control network is introduced and its algebraic form is also presented, which corresponds to a conventional logical control network (i.e., \mu=1). Then it is proved that the optimization technique developed for conventional logical control networks is also applicable to this \mu-memory case. [ABSTRACT FROM AUTHOR]

An algorithm is presented for copying an arbitrarily linked list structure into a block of contiguous storage locations without destroying the original list. Apart from a fixed number of program variables, no auxiliary storage, such as a stack, is used. The algorithm needs no mark bits and operates in linear time. It is shown to be significantly faster than Fisher's algorithm, the fastest previous linear-time algorithm for the same problem. Its speed comes mainly from its efficient list-traversal technique, which folds the processing stack into the structure being built, and from its classification of list cells into nine types, which enables processing operations to be optimized for each type. [ABSTRACT FROM AUTHOR]

Abstract: The goal of designing optimal nearest neighbor classifiers is to maximize classification accuracy while minimizing the sizes of both reference and feature sets. A usual way is to adaptively weight the three objectives as an objective function and then use a single-objective optimization method for achieving this goal. This paper proposes a multi-objective approach to cope with the weight tuning problem for practitioners. A novel intelligent multi-objective evolutionary algorithm IMOEA is utilized to simultaneously edit compact reference and feature sets for nearest neighbor classification. Three comparison studies are designed to evaluate performance of the proposed approach. It is shown empirically that the IMOEA-designed classifiers have high classification accuracy and small sizes of reference and feature sets. Moreover, IMOEA can provide a set of good solutions for practitioners to choose from in a single run. The simulation results indicate that the IMOEA-based approach is an expedient method to design nearest neighbor classifiers, compared with an existing single-objective approach. [Copyright &y& Elsevier]

Differential evolution (DE) was very successful in solving the global continuous optimization problem. It mainly uses the distance and direction information from the current population to guide its further search. Estimation of distribution algorithm (EDA) samples new solutions from a probability model which characterizes the distribution of promising solutions. This paper proposes a combination of DE and EDA (DE/EDA) for the global continuous optimization problem. DE/EDA combines global information extracted by EDA with differential information obtained by DE to create promising solutions. DE/EDA has been compared with the best version of the DE algorithm and an EDA on several commonly utilized test problems. Experimental results demonstrate that DE/EDA outperforms the DE algorithm and the EDA. The effect of the parameters of DE/EDA to its performance is investigated experimentally. [ABSTRACT FROM AUTHOR]

The optimization of real scalar functions of quaternion variables, such as the mean square error or array output power, underpins many practical applications. Solutions typically require the calculation of the gradient and Hessian. However, real functions of quaternion variables are essentially nonanalytic, which are prohibitive to the development of quaternion-valued learning systems. To address this issue, we propose new definitions of quaternion gradient and Hessian, based on the novel generalized Hamilton-real (GHR) calculus, thus making a possible efficient derivation of general optimization algorithms directly in the quaternion field, rather than using the isomorphism with the real domain, as is current practice. In addition, unlike the existing quaternion gradients, the GHR calculus allows for the product and chain rule, and for a one-to-one correspondence of the novel quaternion gradient and Hessian with their real counterparts. Properties of the quaternion gradient and Hessian relevant to numerical applications are also introduced, opening a new avenue of research in quaternion optimization and greatly simplified the derivations of learning algorithms. The proposed GHR calculus is shown to yield the same generic algorithm forms as the corresponding real- and complex-valued algorithms. Advantages of the proposed framework are illuminated over illustrative simulations in quaternion signal processing and neural networks. [ABSTRACT FROM AUTHOR]

We consider a stationary waiting time in a Poisson driven single-server m-node tandem queue with either constant or non-overlapping service times. Each node except for the first one has a finite buffer, and is operated under production blocking. By using (max,+)-algebra, we explicitly express the stationary waiting time at each node as a function of finite buffer capacities. We also address that the explicit expression is separable and monotone decreasing with respect to the buffer sizes. These properties are applied to a buffer allocation problem with probabilistic constraints on stationary waiting times. [ABSTRACT FROM AUTHOR]

We consider deterministic mixed-line flow shop systems that are composed of controllable and uncontrollable machines. Arrival times and completion deadlines of jobs are assumed to be known, and they are processed in the order they arrive at the machines. We model these flow shops as serial networks of queues operating under a non-preemptive first-come-first-served policy, and employ max-plus algebra to characterize the system dynamics. Defining completion-time costs for jobs and service costs at controllable machines, a non-convex optimization problem is formulated where the control variables are the constrained service times at the controllable machines. In order to simplify this optimization problem, under some cost assumptions, we show that no waiting is observed on the optimal sample path at the downstream of the first controllable machine. We also present a method to decompose the optimization problem into convex subproblems. A solution algorithm utilizing these findings is proposed, and a numerical study is presented to evaluate the performance improvement due to this algorithm. [ABSTRACT FROM AUTHOR]

In this technical note, we present a novel approach to robust semidefinite programs, of which coefficient matrices depend polynomially on uncertain parameters. The approach is based on approximation with the sum-of-squares polynomials, but, in contrast to the conventional sum-of- squares approach, the quality of approximation is improved by dividing the parameter region into several subregions. The optimal value of the approximate problem converges to that of the original problem as the resolution of the division becomes finer. An advantage of this approach is that an upper bound on the approximation error can be explicitly obtained in terms of the resolution of the division. A numerical example on polynomial optimization is presented to show usefulness of the present approach. [ABSTRACT FROM AUTHOR]

This note investigates the gap existing between positive polynomials and sum of squares (SOS) of polynomials, which affects several analysis and synthesis tools in control systems based on polynomial SOS relaxations, and about which almost nothing is known. In particular, a matrix characterization of the PNS, that is the positive homogeneous forms that are not SOS, is proposed, which allows to show that any PNS is the vertex of an unbounded cone of PNS. Moreover, a complete parametrization of the set of PNS is introduced. [ABSTRACT FROM AUTHOR]