A Parallel Algorithm for Linear Programs with an Additional Reverse Convex Constraint

A parallel method for globally minimizing a linear program with an additional reverse convex constraint is proposed which combines the outer approximation technique and the cutting plane method. Basicallyp(≤n) processors are used for a problem withnvariables and a globally optimal solution is found effectively in a finite number of steps. Computational results are presented for test problems with a number of variables up to 80 and 63 linear constraints (plus nonnegativity constraints). These results were obtained on a distributed-memory MIMD parallel computer, DELTA, by running both serial and parallel algorithms with double precision. Also, based on 40 randomly generated problems of the same size, with 16 variables and 32 linear constraints (plusx≥ 0), the numerical results from different number processors are reported, including the serial algorithm's.     

Preemptive open shop scheduling with multiprocessors: polynomial cases and applications

This paper addresses a multiprocessor generalization of the preemptive open-shop scheduling problem. The set of processors is partitioned into two groups and the operations of the jobs may require either single processors in either group or simultaneously all processors from the same group. We consider two variants depending on whether preemptions are allowed at any fractional time points or only at integer time points. We reduce the former problem to solving a linear program in strongly polynomial time, while a restricted version of the second problem is solved by rounding techniques. Applications to course scheduling and hypergraph edge coloring are also discussed.     

Newton-conjugate gradient （CG） augmented Lagrangian method for path constrained dynamic process optimization

In this paper, a Newton-conjugate gradient (CG) augmented Lagrangian method is proposed for solving the path constrained dynamic process optimization problems. The path constraints are simplified as a single final time constraint by using a novel constraint aggregation function. Then, a control vector parameterization (CVP) approach is applied to convert the constraints simplified dynamic optimization problem into a nonlinear programming (NLP) problem with inequality constraints. By constructing an augmented Lagrangian function, the inequality constraints are introduced into the augmented objective function, and a box constrained NLP problem is generated. Then, a linear search Newton-CG approach, also known as truncated Newton (TN) approach, is applied to solve the problem. By constructing the Hamiltonian functions of objective and constraint functions, two adjoint systems are generated to calculate the gradients which are needed in the process of NLP solution. Simulation examples demonstrate the effectiveness of the algorithm.     

Finding Total Unimodularity in Optimization Problems Solved by Linear Programs

A popular approach in combinatorial optimization is to model problems as integer linear programs. Ideally, the relaxed linear program would have only integer solutions, which happens for instance when the constraint matrix is totally unimodular. Still, sometimes it is possible to build an integer solution with the same cost from the fractional solution. Examples are two scheduling problems (Baptiste and Schieber, J. Sched. 6(4):395–404, 2003; Brucker and Kravchenko, J. Sched. 11(4):229–237, 2008) and the single disk prefetching/caching problem (Albers et al., J. ACM 47:969–986, 2000). We show that problems such as the three previously mentioned can be separated into two subproblems: (1) finding an optimal feasible set of slots, and (2) assigning the jobs or pages to the slots. It is straigthforward to show that the latter can be solved greedily. We are able to solve the former with a totally unimodular linear program, from which we obtain simple combinatorial algorithms with improved worst case running time.     

An improved heuristic for minimizing makespan among m identical parallel processors

A simple heuristic for minimizing makespan among identical processors assigned independent tasks is presented and explored. The heuristic initially assigns jobs to processors by applying a quick and effective algorithm which at present is commonly applied to this problem. The heuristic then seeks to identify pairs of jobs that may be interchanged between processors to improve the solution. The conditions under which an optimal makespan may be achieved by such an interchange are derived for the case of two processors. The procedure is then extended to three or more processors. Results of 700 randomly generated problems are reported. The heuristic achieved an optimal solution for most of the problems. The worst case performance for the heuristic has not been established; however, evidence is presented that the worst case for the heuristic is considerably smaller than that of algorithms presently used.     

A hybrid evolutionary approach for heterogeneous multiprocessor scheduling

This article investigates the assignment of tasks with interdependencies in a heterogeneous multiprocessor environment; specific to this problem, task execution time varies depending on the nature of the tasks as well as with the processing element assigned. The solution to this heterogeneous multiprocessor scheduling problem involves the optimization of complete task assignments and processing order between the assigned processors to arrive at a minimum makespan, subject to a precedence constraint. To solve an NP-hard combinatorial optimization problem, as is typified by this problem, this paper presents a hybrid evolutionary algorithm that incorporates two local search heuristics, which exploit the intrinsic structure of the solution, as well as through the use of specialized genetic operators to promote exploration of the search space. The effectiveness and contribution of the proposed features are subsequently validated on a set of benchmark problems characterized by different degrees of communication times, task, and processor heterogeneities. Preliminary results from simulations demonstrate the effectiveness of the proposed algorithm in finding useful schedule sets based on the set of new benchmark problems.     

Model validation of multirate systems from time-domain experimentaldata

The model validation problem using time-domain experimental data is studied for multirate linear fractional uncertain models in this note. As a technical tool, the Caratheodory-Fejer (CF) interpolation problem with a nest operator constraint is first investigated. This problem is itself of interest mathematically and has potential applications in addressing other problems in control, signal processing, and circuit theory. A necessary and sufficient solvability condition for this interpolation problem is given. The validation tests are then presented based on this condition and the lifting technique. Tractable convex optimization methods can be used to solve the validation problems     

High Performance Computations for Large Scale Simulations of Subsurface Multiphase Fluid and Heat Flow

TOUGH2 is a widely used reservoir simulator for solving subsurface flow related problems such as nuclear waste geologic isolation, environmental remediation of soil and groundwater contamination, and geothermal reservoir engineering. It solves a set of coupled mass and energy balance equations using a finite volume method. This contribution presents the design and analysis of a parallel version of TOUGH2. The parallel implementation first partitions the unstructured computational domain. For each time step, a set of coupled non-linear equations is solved with Newton iteration. In each Newton step, a Jacobian matrix is calculated and an ill-conditioned non-symmetric linear system is solved using a preconditioned iterative solver. Communication is required for convergence tests and data exchange across partitioning borders. Parallel performance results on Cray T3E-900 are presented for two real application problems arising in the Yucca Mountain nuclear waste site study. The execution time is reduced from 7504 seconds on two processors to 126 seconds on 128 processors for a 2D problem involving 52,752 equations. For a larger 3D problem with 293,928 equations the time decreases from 10,055 seconds on 16 processors to 329 seconds on 512 processors.     

Optimizing the Reliability of Streaming Applications Under Throughput Constraints

Mapping a pipelined application onto a distributed and parallel platform is a challenging problem. The problem becomes even more difficult when multiple optimization criteria are involved, and when the target resources are heterogeneous (processors and communication links) and subject to failures. This paper investigates the problem of mapping pipelined applications, consisting of a linear chain of stages executed in a pipeline way, onto such platforms. The objective is to optimize the reliability under a performance constraint, i.e., while guaranteeing a threshold throughput. In order to increase reliability, we replicate the execution of stages on multiple processors. We compare interval mappings, where the application is partitioned into intervals of consecutive stages, with general mappings, where stages may be partitioned without any constraint, thereby allowing a better usage of processors and communication network capabilities. However, the price to pay for general mappings is a dramatic increase in the problem complexity. We show that computing the period of a given general mapping is an NP-complete problem, and we give polynomial bounds to determine a (conservative) approximated value. On the contrary, the period of an interval mapping obeys a simple formula, and we provide an optimal dynamic programming algorithm for the bi-criteria interval mapping problem on homogeneous platforms. On the more practical side, we design a set of efficient heuristics, and we compare the performance of interval and general mapping strategies through extensive simulations.     

Decentralized control: a stable proper fractional approach

Three decentralized control problems are considered in a fractional setup for two-channel multivariable systems. All three problems are instances of decentralized control or local output feedback problems. The problems are: (i) making the system stabilizable and detectable through the first channel via dynamic output feedback around the second channel, (ii) the first problem with the constraint of internal stability, and (iii) the decentralized stabilization problem. All three problems are equivalent as far as the solvability conditions are concerned. A characterization of all solutions in each case is given. The results apply to a class of systems having fractional representations over an arbitrary principal ideal domain     

An Optimization Approach to Petri Net Monitor Design

This note addresses the problem of enforcing generalized mutual exclusion constraints on a Petri net plant. First, we replace the classical partition of the event set into controllable and uncontrollable events from supervisory control theory, by associating a control and observation cost to each event. This leads naturally to formulate the supervisory control problem as an optimal control problem. Monitor places which enforce the constraint are devised as a solution of an integer linear programming problem whose objective function is expressed in terms of the introduced costs. Second, we consider timed models for which the monitor choice may lead to performance optimization. If the plant net belongs to the class of mono-T-semiflow nets, we present an integer linear fractional programming approach to synthesize the optimal monitor so as to minimize the cycle time lower bound of the closed loop net. For strongly connected marked graphs the cycle time of the closed-loop net can be minimized     

ε-Optimality and duality for multiobjective fractional programming

Using the scalar ε-parametric approach, we establish the Karush-Kuhn-Tucker (which we call KKT) necessary and sufficient conditions for an ε-Pareto optimum of nondifferentiable multiobjective fractional objective functions subject to nondifferentiable convex inequality constraints, linear equality constraints, and abstract constraints. These optimality criteria are utilized as a basis for constructing one duality model with appropriate duality theorems. Subsequently, we employ scalar exact penalty function to transform the multiobjective fractional programming problem to an unconstrained problem. Under this case, we derive the KKT necessary and sufficient conditions without a constraint qualification for ε-Pareto optimality of multiobjective fractional programming.     

Optimal Use of Mixed Task and Data Parallelism for Pipelined Computations

This paper addresses optimal mapping of parallel programs composed of a chain of data parallel tasks onto the processors of a parallel system. The input to the programs is a stream of data sets, each of which is processed in order by the chain of tasks. This computation structure, also referred to as a data parallel pipeline, is common in several application domains, including digital signal processing, image processing, and computer vision. The parameters of the performance for such stream processing are latency (the time to process an individual data set) and throughput (the aggregate rate at which data sets are processed). These two criteria are distinct since multiple data sets can be pipelined or processed in parallel. The central contribution of this research is a new algorithm to determine a processor mapping for a chain of tasks that optimizes latency in the presence of a throughput constraint. We also discuss how this algorithm can be applied to solve the converse problem of optimizing throughput with a latency constraint. The problem formulation uses a general and realistic model of intertask communication and addresses the entire problem of mapping, which includes clustering tasks into modules, assigning of processors to modules, and possible replicating of modules. The main algorithms are based on dynamic programming and their execution time complexity is polynomial in the number of processors and tasks. The entire framework is implemented as an automatic mapping tool in the Fx parallelizing compiler for a dialect of High Performance Fortran.     

Constrained Fractional Variational Problems of Variable Order

Isoperimetric problems consist in minimizing or maximizing a cost functional subject to an integral constraint. In this work, we present two fractional isoperimetric problems where the Lagrangian depends on a combined Caputo derivative of variable fractional order and we present a new variational problem subject to a holonomic constraint. We establish necessary optimality conditions in order to determine the minimizers of the fractional problems. The terminal point in the cost integral, as well as the terminal state, are considered to be free, and we obtain corresponding natural boundary conditions.      

Constraints and universal algebra

In this paper we explore the links between constraint satisfaction problems and universal algebra. We show that a constraint satisfaction problem instance can be viewed as a pair of relational structures, and the solutions to the problem are then the structure preserving mappings between these two relational structures. We give a number of examples to illustrate how this framework can be used to express a wide variety of combinatorial problems, many of which are not generally considered as constraint satisfaction problems. We also show that certain key aspects of the mathematical structure of constraint satisfaction problems can be precisely described in terms of the notion of a Galois connection, which is a standard notion of universal algebra. Using this result, we obtain an algebraic characterisation of the property of minimality in a constraint satisfaction problem. We also obtain a similar algebraic criterion for determining whether or not a given set of solutions can be expressed by a constraint satisfaction problem with a given structure, or a given set of allowed constraint types. This revised version was published online in June 2006 with corrections to the Cover Date.     

Two-level iterated local search for WDM network design problem with traffic grooming

Two important problems arise in WDM network planning: network design to minimize the operation cost and traffic grooming to maximize the usage of the high capacity channels. In practice, however, these two problems are usually simultaneously tackled, denoted as the network design problem with traffic grooming (NDG). In this paper, a mathematical formulation of the NDG problem is first presented. Then, this paper proposes a new metaheuristic algorithm based on two-level iterated local search (TL-ILS) to solve the NDG problem, where a novel tree search based neighborhood construction and a fast evaluation method are proposed, which not only enhance the algorithm's search efficiency but also provide a new perspective in designing neighborhoods for problems with graph structures. Our algorithm is tested on a set of benchmarks generated according to real application scenarios. We also propose a strengthening formulation of the original problem and a method to obtain the lower bound of the NDG problem. Computational results in comparison with the commercial software CPLEX and the lower bounds show the effectiveness of the proposed algorithm.     

一种针对动态部分可重构SoC软硬件划分的高效MILP模型

异构片上系统(System-on-Chip,SoC)在同一芯片上集成了多种类型的处理器,在处理能力、尺寸、重量、功耗等各方面有较大优势,因此在很多领域得到了应用。具有动态部分可重构特性的SoC(Dynamic Partial Reconfigurability SoC,DPR-SoC)是异构SoC的一种重要类型,这种系统兼具了软件的灵活性和硬件的高效性。此类系统的设计通常涉及到软硬件协同问题,其中如何进行应用的软硬件划分是保证系统实时性的关键技术。DPR-SoC中的软硬件划分问题可归类为组合优化问题,问题目标是获得调度长度最短的调度方案,包括任务映射、排序和定时。混合整数线性规划(Mixed Integer Linear Programming,MILP)是求解组合优化问题的一种有效方法;然而,将具体问题建模为MILP模型是求解问题的关键一环,不同建模方式对问题求解时间有重要影响。已有针对DPR-SoC软硬件划分问题的MILP模型存在大量变量和约束方程,对问题求解时间产生了不利影响;此外,其假设条件过多,使得求解结果与实际应用不符。针对这些问题,提出了一种新颖的MILP模型,其极大地降低了模型复杂度,提高了求解结果与实际应用的符合度。将应用建模成DAG图,并使用整数线性规划求解工具对问题进行求解。大量求解结果表明,新的模型能够有效地降低模型复杂度,缩短求解时间;并且随着问题规模的增大,所提模型在求解时间上的优势表现得更加显著。     

Some numerical methods for limit analysis in continuum mechanics

A class of nonlinear optimization problems arises in various fields of continuum physics with a common mathematical structure. The problems are governed by a linear partial differential equation subjected to a quadratic inequality constraint. A linear objective function is minimized or maximized over the set of feasible solutions. The problem is approximated in a finite dimensional space in the form of a special nonlinear programming problem. Several techniques for solving the structured nonlinear programming problem are presented and compared with computational results for a sample problem in plasticity.     

Scheduling divisible loads on heterogeneous linear daisy chain networks with arbitrary processor release times

The problem of distributing and processing a divisible load in a heterogeneous linear network of processors with arbitrary processors release times is considered. A divisible load is very large in size and has computationally intensive CPU requirements. Further, it has the property that the load can be partitioned arbitrarily into any number of portions and can be scheduled onto processors independently for computation. The load is assumed to arrive at one of the farthest end processors, referred to as boundary processors, for processing. The processors in the network are assumed to have nonzero release times, i.e., the time instants from which the processors are available for processing the divisible load. Our objective is to design a load distribution strategy by taking into account the release times of the processors in such a way that the entire processing time of the load is a minimum. We consider two generic cases in which all processors have identical release times and when all processors have arbitrary release times. We adopt both the single and multiinstallment strategies proposed in the divisible load scheduling literature in our design of load distribution strategies, wherever necessary, to achieve a minimum processing time. Finally, when optimal strategies cannot be realized, we propose two heuristic strategies, one for the identical case, and the other for nonidentical release times case, respectively. Several conditions are derived to determine whether or not optimal load distribution exists and illustrative examples are provided for the ease of understanding.     

Design and performance evaluation of load distribution strategies for multiple divisible loads on heterogeneous linear daisy chain networks

In this paper, we consider the problem of scheduling multiple divisible loads on heterogeneous linear daisy chain networks. Our objective is to design a load distribution strategy such that the total processing time of a set of loads is minimized. We assume that the set of loads are resident in one of the farthest end processors, which has a scheduler that will distribute the load to the other processors in the network. When distributing a load from the set, the distribution pattern of the previous load has to be taken into consideration to ensure that no processors are left idle and there are no collisions in the communication links. We design single and multi-installments strategies to achieve the above objective. We derive certain important conditions to determine whether an optimum solution exists. We propose two heuristic strategies when an optimum solution is unattainable. Using all the above strategies, we conduct four different simulation experiments to track the performance of strategies under several real-life situations. We conducted four different simulation experiments based on the two heuristic strategies to identify the best combination suitable for our multiple-loads distribution strategy. We also run simulations for a homogeneous system to quantify the performance under 3 different policies, that is, when the loads are (a) unsorted, (b) sorted with smallest load first (SLF) and (c) sorted with largest load first (LLF). A detailed analysis of the simulation results is presented and based on these, recommendations are made for the choice of strategies. Finally, we compare the performance of a single-load distribution strategy against the multiple-loads distribution strategy designed in this paper to quantify the exact performance gain that can be achieved. Illustrative examples are also provided for ease of understanding.