Persistency of linear programming relaxations for the stable set problem

We investigate the problem of optimally placing limited-range wavelength converters at a subset of nodes in wavelength-routed wavelength-division-multiplexing (WDM) networks. We consider two different aspects of the converter-placement problem: 1) the placement minimizing the network-wide blocking probability; and 2) the placement minimizing the number of wavelength-convertible nodes to meet the performance constraints. We present a binary (0-1) linear program (BLP) formulation in which the end-to-end blocking probability is expressed as a linear function of converter locations so that standard linear program (LP) optimization packages can be employed to obtain the optimal solution of the problems. We also present a new analytical model for estimating the end-to-end blocking probabilities. Experiments have been conducted over four network topologies, including 19-node European optical network (EON), 24-node USA backbone network (UBN), 32-node HYPERCUBE, and 36-node MESH-TORUS. We demonstrate that the optimal solutions of the converter-placement problems can be obtained within a reasonable computation time.

The study deals with the multi-choice mathematical programming problem, where the right hand side of the constraints is multi-choice in nature. However, the problem of multi-choice linear programming cannot be solved directly by standard linear or nonlinear programming techniques. The aim of this paper is to transform such problems to a standard mathematical linear programming problem. For each constraint, exactly one parameter value is selected out of a multiple number of parameter values. This process of selection can be established in different ways. In this paper, we present a new simple technique enabling us to handle such problem as a mixed integer linear programming problem and consequently solve them by using standard linear programming software. Our main aim depends on inserting a specific number of binary variables and using them to construct a linear combination which gives just one parameter among the multiple choice values for each choice of the values of the binary variables. A numerical example is presented to illustrate our analysis.

We formally derive the standard deterministic linear program (LP) for bid-price control by making an affine functional approximation to the optimal dynamic programming value function. This affine functional approximation gives rise to a new LP that yields tighter bounds than the standard LP. Whereas the standard LP computes static bid prices, our LP computes a time trajectory of bid prices. We show that there exist dynamic bid prices, optimal for the LP, that are individually monotone with respect to time. We provide a column generation procedure for solving the LP within a desired optimality tolerance, and present numerical results on computational and economic performance.

We study the ATSP (Asymmetric Traveling Salesman Problem), and our focus is on negative results in the framework of the Sherali-Adams (SA) Lift and Project method. Our main result pertains to the standard LP (linear programming) relaxation of ATSP, due to Dantzig, Fulkerson, and Johnson. For any fixed integer t≥0 and small e, 0<e≪1, there exists a digraph G on ν=ν(t,e)=O(t/e) vertices such that the integrality ratio for level t of the SA system starting with the standard LP on G is ≥ 1+ 1-&epsilon/2t+3 ≈ 4/3, 6/5, 8/7, …. Thus, in terms of the input size, the result holds for any t=0,1,…,Θ(ν) levels. Our key contribution is to identify a structural property of digraphs that allows us to construct fractional feasible solutions for any level t of the SA system starting from the standard LP. Our hard instances are simple and satisfy the structural property. There is a further relaxation of the standard LP called the balanced LP, and our methods simplify considerably when the starting LP for the SA system is the balanced LP; in particular, the relevant structural property (of digraphs) simplifies such that it is satisfied by the digraphs given by the well-known construction of Charikar, Goemans and Karloff (CGK). Consequently, the CGK digraphs serve as hard instances, and we obtain an integrality ratio of $1 +\frac{1-\epsilon}{t+1}$ for any level t of the SA system, where 0<e≪1 and the number of vertices is ν(t,e)=O((t/e)(t/e)). Also, our results for the standard LP extend to the path ATSP (find a min cost Hamiltonian dipath from a given source vertex to a given sink vertex).

Connectionist networks for pivot selection in linear programming

Individual or cooperation as the producer who produces more than one product always has a problem to determine the amount of each product, as well as the smallest packaging used to generate the maximum revenue but still meet the inventory of existing facilities. If the inventory of facilities and functions form a linear function of income then the problem is called the program a maximu m linear case, where all supplies are limited means of the standard linear program is called maximal. Although the system has available a linear program to solve the problem, the system is only owned by a limited circle, this system also does not provide the smallest packaging option. In general linear programming problem is solved by the simplex method with any number of real output in decimal or fractional format mix. Settlement is not operational because the manufacturers do not always able to make the smallest product packaging in accordance with the unit on completion, to become operational, the results of this settlement may be rounded according to the smallest package selected. The problem of limited manufactures which have a system of linear program application can be overcome by providing a system similar to a web-based applications. This study compiled a web-based application system to solve a standard linear program that generates the maximum revenue that maximum use PHP software, data entered through the form and then taken by the POST method, then set up a matrix, carried out by iterating between the rows in the matrix operations that met the criteria optimal. By ch oosing the smallest packaging which is used 0.25 units or 0.50 units, the system check the number of products that have the smallest rounding according to the package selected, then the system prepare and present the inventory and revenue report. Keywords: Simplex metho; The smallest packaging; Web-based

An important problem in WDM network design is to construct a logical topology and determine an optimal routing over that topology. Mixed integer linear program (MILP) formulations to generate optimal solutions for this problem are computationally intractable, even for moderate sized networks. A standard approach is to decouple the problem of logical topology design and the problem of routing on this logical topology. Heuristics for finding the logical topology exist and a straightforward linear program (LP), based on the node-arc formulation is normally used to solve the routing problem over a given logical topology. Such LP formulations become computationally infeasible for large networks. In this paper, we present a new formulation for optimally routing traffic over a given logical topology. Our formulation is based on the arc-chain representation and minimizes the congestion of the network. We have used an implicit column generation technique and exploited the special generalized upper bounding structure of our formulation to efficiently solve the problem.

Linear programming formulation for non-stationary, finite-horizon Markov decision process models

Recent Advances in Mobile Robotics 128 gradient field Unfortunately, path planning approach based on the function family of potential field cannot obtain the optimal objective function which, in turn, cannot guarantee the desired optimal trajectories. Additionally, some of them, for example, potential panel method and harmonic potential function, can plan a path for an autonomous vehicle, but the computing burdens is huge and real time performance hardly satisfies the practical requirement. Inspired by biological intelligence, many approaches, such as ant colony optimization (ACO) They mostly rely on the stochastic searching, known as non-deterministic algorithm. These methods will eventually find an optimal solution, but no estimate on the time of convergence can be given. Thus, it may take long even infinite time to find the best solution. Furthermore, all of them have a great number of parameters to tune and that is never an easy job particularly when users are short of prior knowledge. Some comparisons Another kind of method is mathematic programming. Based on the relative velocity coordinates (RVCs), linear programming (LP) and mixed integer linear programming (MILP) can be employed for path planning problem. For example, Wang et al. converted the 2D path planning of an unmanned underwater vehicle to constrained optimization or semiinfinite constrained optimization problem Zu et al. discussed the path planning in 2D and an LP method was proposed for the problem of dynamic target pursuit and obstacle avoidance This method tried to plan the variables of the linear acceleration and the angular acceleration of a ground vehicle. Schouwenaars et al. proposed a MILP formulation with receding horizon strategy where a minimum velocity and a limited turn rate of aircraft are constrained However, these results still focused on the two dimensions. In this chapter, we consider the same problem of TPOA but in three dimensions. To our problem, the uncertain environment has one target or some of them, denoted by G, and many obstacles O that are all velocity-changeable with their moving actions. The aim of the vehicle A is to find an optimal path for pursuing the target while, at the same time, avoiding collision threaten from obstacles. The position and velocity of movers, including the target and obstacles, are assumed known or estimated at current time. To be more specific, it is assumed that the noise contained in the data can be eliminated with certain filters. However, this paper has no intend to discuss the filtering algorithm for achieving the feasible data of on-board sensors. The trajectory from a start to a destination location typically needs to be computed gradually over time in the fashion of receding horizon The target and obstacles, probably static or dynamic, are modeled by spheres having a certain radius for their impact area, and the vehicle is modeled by a mass point. In order to optimize the vehicle's acceleration online, we construct a linear programming model in Cartesian orthogonal coordinates based on relative velocity space This chapter is organized as follows. First, the relative velocity coordinates are introduced in both two dimensions and three dimensions and then the TPOA principles are introduced, including obstacle-avoidance and target-pursuit principles. The formulation of the standard linear programming is mathematically introduced in Section 3.1, and the path planning model is described in detail in Section 3.2. Simulations of path planning in 2D and 3D are www.intechopen.com Path Planning of Mobile Robot in Relative Velocity Coordinates 129 A, G, O Vehicle, target, obstacle Planning period

Motivated by [12], we provide a framework for studying the size of linear programming formulations as well as semidefinite programming formulations of combinatorial optimization problems without encoding them first as linear programs. This is done via a factorization theorem for the optimization problem itself (and not a specific encoding of such). As a result we define a consistent reduction mechanism that degrades approximation factors in a controlled fashion and which, at the same time, is compatible with approximate linear and semidefinite programming formulations. Moreover, our reduction mechanism is a minor restriction of classical reductions establishing inapproximability in the context of PCP theorems. As a consequence we establish strong linear programming inapproximability (for LPs with a polynomial number of constraints) for several problems that are not 0/1-CSPs: we obtain a 3/2-epsilon inapproximability for Vertex Cover (which is not of the CSP type) answering an open question in [12], we answer a weak version of our sparse graph conjecture posed in [6] showing an inapproximability factor of 1/2+e for bounded degree IndependentSet, and we establish inapproximability of MaxMULTICUT (a non-binary CSP). In the case of SDPs, we obtain relative inapproximability results for these problems.

For many of us, modern-day linear programming (LP) started with the work of George Dantzig in 1947. However, it must be said that many other scientists have also made seminal contributions to the subject, and some would argue that the origins of LP predate Dantzig’s contribution. It is matter open to debate [36]. However, what is not open to debate is Dantzig’s key contribution to LP computation. In contrast to the economists of his time, Dantzig viewed LP not just as a qualitative tool in the analysis of economic phenomena, but as a method that could be used to compute actual answers to specific real-world problems. Consistent with that view, he proposed an algorithm for solving LPs, the simplex algorithm [12]. To this day the simplex algorithm remains a primary computational tool in linear and mixed-integer programming (MIP). In [11] it is reported that the first application of Dantzig’s simplex algorithm to the solution of a non-trivial LP was Laderman’s solution of a 21 constraint, 77 variable instance of the classical Stigler Diet Problem [41]. It is reported that the total computation time was 120 man-days! The first computer implementation of an at-least modestly general version of the simplex algorithm is reported to have been on the SEAC computer at the then National Bureau of Standards [25]. (There were apparently some slightly earlier implementations for dealing with models that were “triangular”, that is, where all the linear systems could be solved by simple addition and subtraction.) Orchard-Hays [35] reports that several small instances having as many as 10 constraints and 20 variables were solved with this implementation. The first systematic development of computer codes for the simplex algorithm began very shortly thereafter at the RAND Corporation in Santa Monica, California. Dantzig’s initial LP work occurred at the Air Force following

Recently, Rothvos established that every linear program (LP) expressing the matching polytope has an exponential number of inequalities (formally, the matching polytope has exponential extension complexity). We generalize this result by deriving strong bounds on the LP inapproximability of the matching problem: for fixed $0 , every $(1 - \varepsilon / n)$ -approximating LP requires an exponential number of inequalities, where $n$ is the number of vertices. This is sharp, given the well-known $\rho $ -approximation of size $O\binom {n}{1/(1 - \rho )}$ provided by the odd-sets of size up to $1/(1 - \rho )$ . Thus, matching is the first problem in $P$ , which does not admit a fully polynomial-size LP relaxation scheme (the LP equivalent of an Fully Polynomial-Time Approximation Scheme), which provides a sharp separation from the polynomial-size LP relaxation scheme obtained, e.g., through constant-sized odd-sets mentioned above. Analyzing the size of LP formulations is equivalent to examining the nonnegative rank of matrices. We study the nonnegative rank through an information–theoretic approach; while it reuses key ideas from Rothvos, the main lower bounding technique is different: we employ the information–theoretic notion of Wyner’s common information used for studying LP formulations. This allows us to analyze the nonnegative rank of perturbations of slack matrices, e.g., the approximations of the matching polytope. It turns out that the high extension complexity for the matching problem stems from the same source of hardness as in the case of the correlation polytope: a direct sum structure.

This paper considers two-dimensional (2-D) retiming, which is the problem of retiming circuits that operate on 2-D signals. We begin by discussing two types of parallelism available in 2-D data processing, which we call inter-iteration parallelism and inter-operation parallelism. We then present two novel techniques for 2-D retiming that can be used to extract inter-operation parallelism. These two techniques are designed to minimize the amount of memory required to implement a 2-D data-flow graph while maintaining a desired clock rate for the circuit. The first technique is based on an integer linear programming (ILP) formulation of the problem, and is called ILP 2-D retiming. This technique considers the entire 2-D retiming problem as a whole, but long central processing unit times are required if the circuit is large. The second technique, called orthogonal 2-D retiming, is a linear programming formulation which is derived by partitioning ILP 2-D retiming into two parts called s- and a-retiming. This technique finds a solution in polynomial time and is much faster than the ILP 2-D retiming technique, but the two sub problems (s- and a-retiming) can give results which are not compatible with one another. To solve this incompatibility problem, a variation of orthogonal 2-D retiming called integer orthogonal 2-D retiming is developed. This technique runs in polynomial time and the s-retiming and a-retiming steps are guaranteed to give compatible results. We show that the techniques presented in this paper can result in memory hardware savings of 50% compared to previously published 2-D retiming techniques.

Among bivariate tail dependence measures, the tail dependence coefficient has emerged as the popular choice. Akin to the correlation matrix, a multivariate dependence measure is constructed using these bivariate measures, and this is referred to in the literature as the tail dependence matrix (TDM). While the problem of determining whether a given d × d matrix is a correlation matrix is of the order O(d3) in complexity, determining if a matrix is a TDM (the realization problem) is believed to be non-polynomial in complexity. Using a linear programming (LP) formulation, we show that the combinatorial structure of the constraints is related to the intractable max-cut problem in a weighted graph. This connection provides an avenue for constructing parametric classes admitting a polynomial in d algorithm for determining membership in its constraint polytope. The complexity of the general realization problem is justifiably of much theoretical interest. Since in practice one resorts to lower dimensional parametrization of the TDMs, we posit that it is rather the complexity of the realization problem restricted to parametric classes of TDMs, and algorithms for it, that are more practically relevant. In this paper, we show how the inherent symmetry and sparsity in a parametrization can be exploited to achieve a significant reduction in the LP formulation, which can lead to polynomial complexity of such realization problems - some parametrizations even resulting in the constraint polytope being independent of d. We also explore the use of a probabilistic viewpoint on TDMs to derive the constraint polytopes.

This paper presents new linear programming techniques for the design of optimal 1D IIR digital filters. the first method is based on a linear minimax criterion and leads to a linear programming approximation problem whose optimal solution is attained by a new fast algorithm. Next, the linear minimax approximation problem is extended and formulated as two linear integer programming problems which permit the design of 1D IIR digital filters with coefficients of finite word length. the first of these methods is formulated as a 0‐1 integer linear programming problem and its optimal solution is attained rapidly by a new algorithm. This method appears to be suitable for the design of high‐order digital filters. the second method is more general and is formulated as a mixed integer linear programming problem. to solve the design problem efficiently, a new integer tree search algorithm is introduced. the feasibility of the proposed algorithms is illustrated with detailed solutions or numerical examples.