Lexicographic perturbation for multiparametric linear programming with applications to control

Optimal control problems for constrained linear systems with a linear cost can be posed as multiparametric linear programs (mpLPs) with a parameter in the cost, or equivalently the right-hand side of the constraints, and solved explicitly offline. Degeneracy occurs when the control input, or optimiser, is non-unique, which can cause chattering of the control input and overlap of the polyhedral regions of the explicit solution. This paper introduces a new and efficient approach to the computation of the solution to a degenerate mpLP with the parameter in the cost. Rather than solve the degenerate problem directly, we solve a lexicographically (symbolically) perturbed version of it that is guaranteed to be non-degenerate. We show that every optimal solution of the perturbed problem is an optimal solution to the original and that the perturbed solution is continuous, unique and defined over a set of non-overlapping polyhedral regions. Furthermore, we introduce a new method for computing the optimal solution in an adjacent region, which is very efficient in all cases and reduces to a single simplex pivot for non-degenerate regions. The proposed algorithm is particularly suited for the calculation of the explicit solution to a class of constrained optimal control problems, since it ensures that the control input is everywhere continuous and unique, thereby removing the danger of chattering in problems with linear costs. The algorithm is compared through example to existing proposals and a significant complexity improvement is demonstrated.     

An efficient search direction for linear programming problems

In this paper, we present an auxiliary algorithm, in terms of the speed of obtaining the optimal solution, that is effective in helping the simplex method for commencing a better initial basic feasible solution. The idea of choosing a direction towards an optimal point presented in this paper is new and easily implemented. From our experiments, the algorithm will release a corner point of the feasible region within few iterative steps, independent of the starting point. The computational results show that after the auxiliary algorithm is adopted as phase I process, the simplex method consistently reduce the number of required iterations by about 40%.Scope and purposeRecent progress in the implementations of simplex and interior point methods as well as advances in computer hardware has extended the capability of linear programming with today's computing technology. It is well known that the solution times for the interior point method improve with problem size. But, experimental evidence suggests that interior point methods dominate simplex-based methods only in the solution of very large scale linear programs. If the problem size is medium, how to combine the best features of these two methods to produce an effective algorithm for solving linear programming problems is still an interesting problem. In this research we present a new effective ε-optimality search direction based on the interior point method to start an initial basic feasible solution near the optimal point for the simplex method.     

退化线性规划的一个新的改进的单纯形方法

本文讨论退化线性规划单纯形方法最优解的判定准则和有限主元规则.首先改进简约价值系数向量,提出线性规划单纯形方法最优解的判定准则.并且利用本文的判定准则给出[3]中定理2.3.5(P.84)的一个新的证明.然后提出一种新的混合有限主元规则,在退化情形下通过对单纯形表使用新的混合有限主元规则进行迭代,可以判断当前退化基本可行解或为最优解或给出下次迭代的主元并且跳出循环.最后给出在一组经典的退化线性规划例子下,改进的单纯形方法好的计算表现.     

Dynamic distributed unicast routing: optimal incremental paths

Developing new mathematical frameworks such as distributed dynamic routing algorithms for constructing optimal incremental paths from a node to another node is an important challenge in data communication networks. These new algorithms can model network resources optimally and increase network performances. A bundle of single routes in a current communication path, which starts from a source node and ends to a destination node, can consist of several successive nodes and links. The Incremental term emphasizes that the number of routes (links and nodes) in a current path can change so that achieving more data rate and optimal efficiency in the network. In this paper, our problem is to add/omit some routes consisting of some nodes and links to/from the current unicast path dynamically and optimally. We call this problem the Optimal Dynamic Distributed Unicast Routing (ODDUR) problem and it is a NP-complete problem. This problem can be formulated as a new type of Linear Programming Problem (LPP) for finding a minimum cost multichannel unicast path, which this path will minimize end-to-end delay and bandwidth consumption along the routes from the source node to the destination node. In this paper, at first a new mathematical framework will be constructed and then this framework will propose the new optimal dynamic distributed unicast routing algorithm for solving our LPP problem. This algorithm will compute an optimal solution for our LPP based on the simplex method and postoptimality computations and will reduce computations and consumed time. Simulation results will show that our new algorithm is more efficient than other available algorithms in terms of utilization of bandwidths and data rate.     

A new hybrid algorithm for the balanced transportation problem

We propose a heuristic-exact hybrid algorithm that consists of a heuristic phase, based on two novel heuristics, followed by an exact phase, based on an adapted Ford–Fulkerson algorithm, to solve the Balanced Transportation Problem (BTP). First, we propose three alternative primal models for the BTP. We also define the concepts of negative sets, negative dual rectangles, and the optimal tableau for the BTP. Next, we explore the relationships between these concepts. We also propose two greedy heuristics, based on a linear programming relaxation of the BTP model, to find some negative sets and negative dual rectangles. These two heuristics turn out to be very efficient and obtain optimal or near-optimal BTP tableaus rapidly, as confirmed by the computational experiments. Then, an adapted Ford–Fulkerson algorithm is presented and used to find an optimal solution. The two important advantages of our adapted Ford–Fulkerson algorithm over the standard Ford–Fulkerson algorithm are more flexibility and efficiency. Extensive computational results show that the growth in run-time of our hybrid algorithm, on average, is approximately a linear function of the BTP size. It has significant advantage over the transportation simplex method and on the largest problem instances it is almost five times faster. A key feature of the proposed algorithm is that it is free of degeneracy and cycling altogether.     

A new version of the Improved Primal Simplex for degenerate linear programs

The Improved Primal Simplex (IPS) algorithm [Elhallaoui I, Metrane A, Desaulniers G, Soumis F. An Improved Primal Simplex algorithm for degenerate linear programs. SIAM Journal of Optimization, submitted for publication] is a dynamic constraint reduction method particularly effective on degenerate linear programs. It is able to achieve a reduction in CPU time of over a factor of three on some problems compared to the commercial implementation of the simplex method CPLEX. We present a number of further improvements and effective parameter choices for IPS. On certain types of degenerate problems, our improvements yield CPU times lower than those of CPLEX by a factor of 12.     

Robust rendezvous for multi-robot system with random node failures: an optimization approach

In this paper, we consider the problem of designing distributed control algorithms to solve the rendezvous problem for multi-robot systems with limited sensing, for situations in which random nodes may fail during execution. We first formulate a distributed solution based upon averaging algorithms that have been reported in the consensus literature. In this case, at each stage of execution a one-step sequential optimal control (i.e., näive greedy algorithm) is used. We propose a distributed stochastic optimal control algorithm that minimizes a mean–variance cost function for each stage, given that the probability distribution for possible node failures is known a priori, as well as a minimax version of the problem when the prior probability distribution is not known. We demonstrate via extensive numerical simulations that our proposed algorithm provides statistically better rendezvous task performance than contemporary algorithms in cases for which failures occur.     

An optimal algorithm for two robots path planning problem on the grid

This paper is a study on the problem of path planning for two robots on a grid. We consider the objective of minimizing the maximum path length which corresponds to minimizing the arrival time of the last robot at its goal position. We propose an optimal algorithm that solves the problem in linear time with respect to the size of the grid. We show that the algorithm is complete; meaning that it is sure to find an optimal solution or report if any does not exist.     

On an exact method for the constrained shortest path problem

The constrained shortest path (CSP) is a well known NP-Hard problem. Besides from its straightforward application as a network problem, the CSP is also used as a building block under column-generation solution methods for crew scheduling and crew rostering problems. We propose an exact solution method for the CSP capable of handling large-scale networks in a reasonable amount of time. We compared our approach with three different state-of-the-art algorithms for the CSP and found optimal solutions on networks with up to 40,000 nodes and 800,000 arcs. We extended the algorithm to effectively solve the auxiliary problems of a multi-activity shift scheduling problem and a bus rapid transit route design problem tackled with column generation. We obtained significant speedups against alternative column generation schemes that solve the auxiliary problem with state-of-the-art commercial (linear) optimizers. We also present a first parallel version of our algorithm that shows promising results.     

Structured sparse matrix-vector multiplication on massively parallel SIMD architectures

We propose an algorithm for the cost-effective evaluation of structured sparse matrix-vector multiplications on massively parallel SIMD computers with distributed memory. Under the assumption that a large percentage of the nonzero entries of the matrix is concentrated on almost full diagonals, a specific data transport problem arises, for which we derive an equivalent graph theoretical formulation. We find an optimal solution of this communication problem in terms of a minimum-cost spanning tree.

We demonstrate the efficiency of our matrix-vector multiplication on a MP-1 with 16,384 processors.     

An LP-based hyperparameter optimization model for language modeling

In order to find hyperparameters for a machine learning model, algorithms such as grid search or random search are used over the space of possible values of the models’ hyperparameters. These search algorithms opt the solution that minimizes a specific cost function. In language models, perplexity is one of the most popular cost functions. In this study, we propose a fractional nonlinear programming model that finds the optimal perplexity value. The special structure of the model allows us to approximate it by a linear programming model that can be solved using the well-known simplex algorithm. To the best of our knowledge, this is the first attempt to use optimization techniques to find perplexity values in the language modeling literature. We apply our model to find hyperparameters of a language model and compare it to the grid search algorithm. Furthermore, we illustrate that it results in lower perplexity values. We perform this experiment on a real-world dataset from SwiftKey to validate our proposed approach.     

A modified algorithm for generalized discriminant analysis

Generalized discriminant analysis (GDA) is an extension of the classical linear discriminant analysis (LDA) from linear domain to a nonlinear domain via the kernel trick. However, in the previous algorithm of GDA, the solutions may suffer from the degenerate eigenvalue problem (i.e., several eigenvectors with the same eigenvalue), which makes them not optimal in terms of the discriminant ability. In this letter, we propose a modified algorithm for GDA (MGDA) to solve this problem. The MGDA method aims to remove the degeneracy of GDA and find the optimal discriminant solutions, which maximize the between-class scatter in the subspace spanned by the degenerate eigenvectors of GDA. Theoretical analysis and experimental results on the ORL face database show that the MGDA method achieves better performance than the GDA method.     

Application of optimal filtering methods for on-line of queueing network states

We show the solution to the optimal filtering problem for states of Markov jump processes by observations of multivariant point processes. A characteristic feature of observations is that their compensators are random linear functions of the system state, and the composite “state–observations” process does not possess the Markov property. The provided optimal filtering estimate is expressed via the solution of some recurrent system of linear differential equations and algebraic relations. We present examples of using theoretical results to construct typical models of real queueing networks. We establish the connections between our new optimal filtering algorithm and classical results of Kalman–Bucy and Wonham. We propose a solution for the problem of estimating the current state of a UDP connection given the observations of video stream.     

Dynamic Multicast in Overlay Networks with Linear Capacity Constraints

In a peer-to-peer overlay network, the phenomenon of multiple overlay links sharing bottleneck physical links leads to correlation of overlay link capacities. We are able to more accurately model the overlay by incorporating these linear capacity constraints (LCCs). We formulate the problem of maximizing bandwidth in overlay multicast using our LCC model. We show that finding a maximum bandwidth multicast tree in an overlay network with LCC is NP-complete. Therefore, an efficient heuristics algorithm is designed to solve the problem. Extensive simulations show that our algorithm is able to construct multicast trees that are optimal or extremely close to optimal, with significantly higher bandwidth than trees formed in overlays with no LCC. Furthermore, we develop a fully distributed algorithm for obtaining near-optimal multicast trees, by means of gossip-based algorithms and a restricted but inherently distributed class of LCC (node-based LCC). We demonstrate that the distributed algorithm converges quickly to the centralized optimal and is highly scalable.     

A numerical algorithm for optimal control of a class of hybrid systems: differential transformation based approach

A novel numerical algorithm based on differential transformation is proposed for optimal control of a class of hybrid systems with a predefined mode sequence. From the necessary conditions for optimality of hybrid systems, the hybrid optimal control problem is first converted into a two-point boundary value problem (TPBVP) with additional transverse conditions at the switching times. Then we propose a differential transformation algorithm for solving the TPBVP which may have discontinuities in the state and/or control input at the switching times. Using differential transformation, the hybrid optimal control problem reduces to a problem of solving a system of algebraic equations. The numerical solution is obtained in the form of a truncated Taylor series. By taking advantage of the special properties of the linear subsystems and a quadratic cost functional, the differential transformation algorithm can be further simplified for the switched linear quadratic optimal control problem. We analyse the error of the numerical solution computed by the differential transformation algorithm and some computational aspects are also discussed. The performance of the differential transformation algorithm is demonstrated through illustrative examples. The differential transformation algorithm has been shown to be simple to be implemented and computationally efficient.     

Degenerate problem of stabilization of a specified trajectory

Consideration is given to the problem of stabilization of a specified (supporting) mode as a degenerate problem of minimization of the root-mean-square deviation from it where there are no deviations of control actions. This formulation makes it possible to completely use available limited control resources for stabilization. We propose a method of approximate optimal synthesis in the neighborhood of a turnpike manifold that is obtained as a solution in the form of optimal synthesis of linearly quadratic problem with unbounded linear control. The investigation is performed for a linearized discrete model of the controlled system; however, the obtained stabilizing control can be applied in the initial nonlinear system directly or after the correction by regular iterations.     

Interprocessor communication with limited memory

Many parallel applications require periodic redistribution of workloads and associated data. In a distributed memory computer, this redistribution can be difficult if limited memory is available for receiving messages. We propose a model for optimizing the exchange of messages under such circumstances which we call the minimum phase remapping problem. We first show that the problem is NP-complete, and then analyze several methodologies for addressing it. First, we show how the problem can be phrased as an instance of multicommodity flow. Next, we study a continuous approximation to the problem. We show that this continuous approximation has a solution which requires at most two more phases than the optimal discrete solution, but the question of how to consistently obtain a good discrete solution from the continuous problem remains open. We also devise a simple and practical approximation algorithm for the problem with a bound of 1.5 times the optimal number of phases. We also present an empirical study of variations of our algorithms which indicate that our approaches are quite practical.     

Sensor deployment optimization methods to achieve both coverage and connectivity in wireless sensor networks

In this study, we aim to cover a sensing area by deploying a minimum number of wireless sensors while maintaining the connectivity between the deployed sensors. The problem may be reduced to a two-dimensional critical coverage problem which is an NP-Complete problem. We develop an integer linear programming model to solve the problem optimally. We also propose a local search (LS) algorithm and a genetic algorithm (GA) as approximate methods. We verify by computational experiments that the integer linear model, using Cplex, is able to provide an optimal solution of all our small and medium size problems. We also show that the proposed methods outperform some regular sensor deployment patterns.     

Maximizing network lifetime based on transmission range adjustment in wireless sensor networks

In a wireless sensor network (WSN), the unbalanced distribution of communication loads often causes the problem of energy hole, which means the energy of the nodes in the hole region will be exhausted sooner than the nodes in other regions. This is a key factor which affects the lifetime of the networks. In this paper we propose an improved corona model with levels for analyzing sensors with adjustable transmission ranges in a WSN with circular multi-hop deployment (modeled as concentric coronas). Based on the model we consider that the right transmission ranges of sensors in each corona is the decision factor for optimizing the network lifetime after nodes deployment. We prove that searching optimal transmission ranges of sensors among all coronas is a multi-objective optimization problem (MOP), which is NP hard. Therefore, we propose a centralized algorithm and a distributed algorithm for assigning the transmission ranges of sensors in each corona for different node distributions. The two algorithms can not only reduce the searching complexity but also obtain results approximated to the optimal solution. Furthermore, the simulation results of our solutions indicate that the network lifetime approximates to that ensured by the optimal under both uniform and non-uniform node distribution.     

网格计算中费用约束的最优时间调度算法

在网格资源处理速度和资源价格异构的网格环境下,讨论基于用户费用约束的最优时间调度问题,提出一种相应的调度算法,将该任务调度问题转化为线性规划问题,采用单纯形算法获得近似最优解,从而获得费用约束下资源的最优执行时间以及该任务的最小完成时间。仿真结果表明,该算法的性能优于其他同类算法。