The location problem of interconnected facilities on the line with forbidden gaps is considered. The located facilities are connected among themselves and with gaps. Location in forbidden gaps is not allowed. It is need to minimize the total cost of connections between facilities and between facilities and gaps. It is known that the initial continuous problem is reduced to series discrete subproblems of smaller dimension. In this paper the definition of the local optimum of the problem is introduced. In order that to obtain the local optimum it is necessary to solve some subproblems. The variants of lower bounds of the goal function of the subproblems are proposed. The bounds can be used in the branch and bounds algorithm for solving the subproblems.
We consider a routing problem with constraints. To solve this problem, we employ a variant of the dynamic programming method, where the significant part (that is, the part that matters in view of precedence constraints) of the Bellman function is calculated by means of an independent calculations scheme. We propose a parallel implementation of the algorithm for a supercomputer, where the construction of position space layers for the hypothetical processors is conducted with use of discrete dynamic systems' apparatus.
We consider the strongly NP-hard problem of partitioning a set of Euclidean points into two clusters so as to minimize the sum (over both clusters) of the weighted sum of the squared intracluster distances from the elements of the clusters to their centers. The weights of sums are the cardinalities of the clusters. The center of one of the clusters is given as input, while the center of the other cluster is unknown and determined as the geometric center (centroid), i.e. the average value over all points in the cluster. We analyze the variant of the problem with cardinality constraints. We present an approximation algorithm for the problem and prove that it is a fully polynomial-time approximation scheme when the space dimension is bounded by a constant.
Energy efficiency has become a crucial issue in Computer Science. New hardware and system-based approaches are explored for saving energy in portable battery-operated devices, personal computers, or large server farms. The main mechanisms that have been developed for saving energy are the ability of transitioning the device among multiple power states, and the use of dynamic voltage scaling (speed scaling).
The article deals with the modern approaches to the study of physical and mechanical properties of materials. A brief description is given on the express method for metal-cutting tool health assessment that has been developed within the framework of this approach. The paper suggests ways to improve the performance of by modeling the properties of materials. It presents the methodology for assessing the tool material performance of coated surfaces by acoustic
This paper considers a network comprised of parallel routes with the Bureau of Public Road (BPR) latency function and suggests an optimal distribution method for incoming traffic flow. The authors analytically derive a system of equations defining the optimal distribution of the incoming flow with minimum social costs, as well as a corresponding system of equations for the Wardrop equilibrium in this network. In particular, the Wardrop equilibrium is applied to the competition model with rational consumers who use the carriers with minimal cost, where cost is equal to the price for service plus the waiting time for the service. Finally, the social costs under the equilibrium and under the optimal distribution are compared. It is shown that the price of anarchy can be infinitely large in the model with strategic pricing.
In Euclidean plane geometry, Apollonius’ problem is to construct a circle in a plane that is tangent to three given circles. We will use a solution to this ancient problem to solve several versions of the following geometric optimization problem. Given is a set of customers located in the plane, each having a demand for a product and a budget. A customer is satisfied if her total, travel and purchase, costs do not exceed her budget. The task is to determine location of production facilities in the plane and one price for the product such that the revenue generated from the satisfied customers is maximized.
We consider a fractional 0-1 programming problem arising in manufacturing. The problem consists in clustering of machines together with parts processed on these machines into manufacturing cells so that intra-cell processing of parts is maximized and inter-cell movement is minimized. This problem is called Cell Formation Problem (CFP) and it is an NP-hard optimization problem with Boolean variables and constraints and with a fractional objective function. Because of its high computational complexity there are a lot of heuristics developed for it. In this paper we suggest a branch and bound algorithm which provides exact solutions for the CFP with a variable number of cells and grouping efficacy objective function. This algorithm finds optimal solutions for 21 of the 35 popular benchmark instances from literature and for the remaining 14 instances it finds good solutions close to the best known.
We consider stochastic programming problems with probabilistic and quantile objective functions. The original distribution of the random variable is replaced by a discrete one. We thus consider a sequence of problems with discrete distributions. We suggest conditions, which guarantee that the sequence of optimal strategies converges to an optimal strategy of the original problem. We consider the case of a symmetrical distribution, the case of the loss function increasing in the random variable, and the case of the loss function increasing in the optimization strategy.
We suggest a partial linearization method for network equilibrium problems with elastic demands, which can be set-valued in general. The main element of this method is a partially linearized auxiliary problem. We propose a simple solution method for the auxiliary problem, which is based on optimality conditions. This method can be viewed as alternative to the conditional gradient method for the single-valued case. Some results of preliminary calculations which confirm efficiency of the new method are also presented.
The separating plane method with additional clippings and stockpiling (SPACLIP-S) for nonsmooth optimization is proposed in this paper. Both the theoretical and experimental investigations of the method showed that this method is efficient and widely applicable to nonsmooth optimization problems with convex objective functions. Computational experiments demonstrated a rather high performance of SPACLIP-S when applied to the Web page ranking problem. Web page ranking approach is widely used by search engines such as Google and Yandex to order Web pages. Page ranking problem is one of the most important problems in information retrivial due to wide range of applications. In this paper an iterative regularization method with a new penalty function for solving PageRank problem is also presented. Finally, experimental results of comparison of SPACLIP-S and other algorithms for solving test PageRank problems are provided.
in this paper the speed gradient method is applied to design an adjustment algorithm for parameters of neural network controller. Local quadratic criterion expresses generalized error of desired trajectory tracking. Continuous adjustment laws for neural network parameters and their discrete analogies are derived on base of speed gradient method. To illustrate an approach, the mathematical model of underwater robot is taken. Numerical experiments had confirmed.
Linear superiorization (abbreviated: LinSup) considers linear programming (LP) problems wherein the constraints as well as the objective function are linear. It allows to steer the iterates of a feasibilityseeking iterative process toward feasible points that have lower (not necessarily minimal) values of the objective function than points that would have been reached by the same feasiblity-seeking iterative process without superiorization. Using a feasibility-seeking iterative process that converges even if the linear feasible set is empty, LinSup generates an iterative sequence that converges to a point that minimizes a proximity function which measures the linear constraints violation. In addition, due to LinSup's repeated objective function reduction steps such a point will most probably have a reduced objective function value. We present an exploratory experimental result that illustrates the behavior of LinSup on an infeasible LP problem.
Two competitors, a Leader and a Follower, are sequentially creating their hub and spoke networks to attract customers in a market where prices have fixed markups. Each competitor wants to maximize his profit, rather than a market share. Demand is split according to the logit model. The goal is to find the optimal hub and spoke topology for the Leader. We represent this Stackelberg game as a nonlinear mixed-integer bi-level optimisation problem and show how to reformulate the Follower’s problem as a mixed-integer linear program. Exploiting this reformulation, we solve instances based on a synthetic data using the alternating heuristic as a solution approach. Computational results are thoroughly discussed, consequently providing some managerial insights.
This study intends to help manufacturers that use flexible job shops improve performance of due date assignment, that is, setting delivery times to jobs that arrive dynamically. High performing due date assignment enables achieving on-time delivery and quick response of delivery time to customer orders. Traditional methods for due date assignment are predefined equations that estimate the duration of making a product in the production system. Such equations are sufficient for relatively simple systems such as single machine shops, but are not very high in accuracy for complex systems such as flexible job shops. To improve due date assignment for such systems, we propose a more flexible method that uses distributed learning to learn the remaining time of a job inside the system. We let each workstation in the production shop be a distributed unit that updates its local queuing time and interacts with other units to provide the total remaining time of a job. We carry out extensive computational experiments to evaluate performance of the proposed method, and the results show that it outperforms two advanced equational methods in terms of both accuracy of estimation and stability in performance.
We consider the problem of partitioning a finite sequence of points in Euclidean space into a given number of clusters (subsequences) minimizing the sum of squared distances between cluster elements and the corresponding cluster centers. It is assumed that the center of one of the desired clusters is the origin, while the centers of the other clusters are unknown and determined as the mean values over clusters elements. Additionally, there are a few structural restrictions on the elements of clusters with unknown centers: (1) clusters form non-overlapping subsequences of the input sequence, (2) the difference between two consecutive indices is bounded from below and above by prescribed constants, and (3) the total number of elements in these clusters is given as an input. It is shown that the problem is strongly NP-hard. A 2-approximation algorithm which runs in polynomial time for a fixed number of clusters is proposed for this problem.
When playing certain specific classes of no-regret algorithms such as multiplicative updates and replicator dynamics in atomic congestion games, some previous convergence analyses were done with the standard Rosenthal potential function in terms of mixed strategy profiles (i.e., probability distributions on atomic flows), which could be non-convex. In several other works, the convergence, when playing the mirror-descent algorithm (a more general family of no-regret algorithms including multiplicative updates, gradient descents, etc.), was guaranteed with a convex potential function in terms of nonatomic flows as an approximation of the Rosenthal one. The convexity of the potential function provides convenience for analysis. One may wonder if the convergence of mirror descents can still be guaranteed directly with the non-convex Rosenthal potential function. In this paper, we answer the question affirmatively for discrete-time generalized mirror descents with the smoothness property (similarly adopted in many previous works for congestion games and markets) and for continuous-time generalized mirror descents with the separability of regularization functions.
In this paper we proposed a modified Lagrangian functional for obstacle problem, investigated its properties. Then we construct Uzawa method for finding a saddle point, proved the convergence theorems. Some numerical examples are provided.
The paper is devoted to upper bounds on the expected first hitting time of the sets of optimal solutions for non-elitist genetic algorithms with very high selection pressure. The results of this paper extend the range of situations where the upper bounds on the expected runtime are known for genetic algorithms and apply, in particular, to the Canonical Genetic Algorithm. The obtained bounds do not require the probability of fitness-decreasing mutation to be bounded by a constant less than one.
We study linear programming with box constraints on all variables (in addition to other linear constraints). Following research of Fujishige et al., we consider this problem as finding an intersection between a line and a specially constructed zonotope. This approach allows us to explore a variety of projection-type optimization algorithms, where we do not need to rely on the solution of linear equations.
