A Dual Simplex Algorithm for Piecewise-Linear Programming

This paper presents a dual piecewise-linear simplex algorithm for minimizing convex separable piecewise-linear functions subject to linear constraints. It is an extension of Fourier's work on piecewise-linear programming to the dual piecewise-linear simplex algorithm. This algorithm has advantages over indirect methods which solve equivalent linear programs augmented by additional variables and/or constraints. Computational experience is presented which demonstrates the efficiency these advantages contribute.     

A constraint programming approach for the resource-constrained project scheduling problem

A “pure” Constraint Programming approach for the Resource-Constrained Project Scheduling Problem (RCPSP) is presented. Our basic idea was to substitute the resource constraints by a set of “sub-constraints” generated as needed. Each of these sub-constraints corresponds to a set of tasks that cannot be executed together without violating one of the resource constraints. A filtering algorithm for these sub-constraints has been developed. When applied to the initial resource constraints together with known filtering algorithms, this new filtering algorithm provides very good numerical results.     

Advantages of simplicial partitioning for Lipschitz optimization problems with linear constraints

The well known DIRECT (DIviding RECTangles) algorithm for global optimization requires bound constraints on variables and does not naturally address additional linear or nonlinear constraints. A feasible region defined by linear constraints may be covered by simplices, therefore simplicial partitioning may tackle linear constraints in a very subtle way. In this paper we demonstrate this advantage of simplicial partitioning by applying a recently proposed deterministic simplicial partitions based DISIMPL algorithm for optimization problems defined by general linear constraints (Lc-DISIMPL). An extensive experimental investigation reveals advantages of this approach to such problems comparing with different constraint-handling methods, proposed for use with DIRECT. Furthermore the Lc-DISIMPL algorithm gives very competitive results compared to a derivative-free particle swarm algorithm (PSwarm) which was previously shown to give very promising results. Moreover, DISIMPL guarantees the convergence to the global solution, whereas the PSwarm algorithm sometimes fails to converge to the global minimum.     

Local search and suffix tree for car-sequencing problem with colors

This paper describes an approximation solution method for the car sequencing problem with colors. Firstly, we study the optimality of problems with a single ratio constraint. This study also introduces a data structure for efficient calculation of the penalties related to ratio constraints. We describe the constructive greedy algorithm and variable neighborhood search adjusted for the problem with colors. Tabu metaheuristic is used to improve the results obtained by VNS. We then represent the cars with their constraints as letters over an alphabet and apply the algorithm to spell the motifs in order to improve the number of batch colors without decreasing the costs associated to the set of ratio constraints. The algorithm achieves 19 out of the 64 best results for instance sets A and B. These instances are the reference instances for Challenge ROADEF.     

A least-distance programming procedure for minimization problems under linear constraints

In this paper, an algorithm is developed for solving a nonlinear programming problem with linear contraints. The algorithm performs two major computations. First, the search vector is determined by projecting the negative gradient of the objective function on a polyhedral set defined in terms of the gradients of the equality constraints and the near binding inequality constraints. This least-distance program is solved by Lemke's complementary pivoting algorithm after eliminating the equality constraints using Cholesky's factorization. The second major calculation determines a stepsize by first computing an estimate based on quadratic approximation of the function and then finalizing the stepsize using Armijo's inexact line search. It is shown that any accumulation point of the algorithm is a Kuhn-Tucker point. Furthermore, it is shown that, if an accumulation point satisfies the second-order sufficiency optimality conditions, then the whole sequence of iterates converges to that point. Computational testing of the algorithm is presented.     

A method of linearizations for linearly constrained nonconvex nonsmooth minimization

A readily implementable algorithm is given for minimizing a (possibly nondifferentiable and nonconvex) locally Lipschitz continuous functionf subject to linear constraints. At each iteration a polyhedral approximation tof is constructed from a few previously computed subgradients and an aggregate subgradient, which accumulates the past subgradient information. This aproximation and the linear constraints generate constraints in the search direction finding subproblem that is a quadratic programming problem. Then a stepsize is found by an approximate line search. All the algorithm's accumulation points are stationary. Moreover, the algorithm converges whenf happens to be convex.     

约束装箱问题的混合遗传算法求解

本将最佳适应法和遗传算法相结合，提出了一种新的启发式混合遗传算法对具有时间约束的装箱问题进行求解，给出了具体的算法步骤，试算结果表明基于启发式算法的混合遗传算法适合于求解各种约束条件下的大规模装箱问题。     

A new constrained fixed-point algorithm for ordering independent components

Independent component analysis (ICA) aims to recover a set of unknown mutually independent components (ICs) from their observed mixtures without knowledge of the mixing coefficients. In the classical ICA model there exists ICs’ indeterminacy on permutation and dilation. Constrained ICA is one of methods for solving this problem through introducing constraints into the classical ICA model. In this paper we first present a new constrained ICA model which composed of three parts: a maximum likelihood criterion as an objective function, statistical measures as inequality constraints and the normalization of demixing matrix as equality constraints. Next, we incorporate the new fixed-point (newFP) algorithm into this constrained ICA model to construct a new constrained fixed-point algorithm. Computation simulations on synthesized signals and speech signals demonstrate that this combination both can eliminate ICs’ indeterminacy to a certain extent, and can provide better performance. Moreover, comparison results with the existing algorithm verify the efficiency of our new algorithm furthermore, and show that it is more simple to implement than the existing algorithm due to its advantage of not using the learning rate. Finally, this new algorithm is also applied for the real-world fetal ECG data, experiment results further indicate the efficiency of the new constrained fixed-point algorithm.     

A modified algorithm for solving the proposed models by Ghodousian and Khorram and Khorram and Ghodousian

In this paper, we focus on the proposed algorithm for optimizing the linear function with fuzzy relation equation constraints regarding max-prod composition that it has been proposed by Ghodousian and Khorram [A. Ghodousian, E. Khorram, An algorithm for optimizing the linear function with fuzzy relation equation constraints regarding max-prod composition, Appl. Math. Comput. 178 (2006) 502–509]. Firstly, we show that the algorithm may not lead to the optimal solution in some cases. Secondly, we propose a new algorithm for solving the presented model by Ghodousian and Khorram (2006), as mentioned above. In fact, it modifies the presented algorithm in the Ghodousian and Khorram’s paper. Also, this algorithm is extended to the presented model by Khorram and Ghodousian [E. Khorram, A. Ghodousian, Linear objective function optimization with fuzzy relation equation constraints regarding max-av composition, Appl. Math. Comput. 173 (2006) 872–886.] with max-av composition. Finally, some numerical examples are given for illustrating the purposes.     

A tolerant algorithm for linearly constrained optimization calculations

Two extreme techniques when choosing a search direction in a linearly constrained optimization calculation are to take account of all the constraints or to use an active set method that satisfies selected constraints as equations, the remaining constraints being ignored. We prefer an intermediate method that treats all inequality constraints with small residuals as inequalities with zero right hand sides and that disregards the other inequality conditions. Thus the step along the search direction is not restricted by any constraints with small residuals, which can help efficiency greatly, particularly when some constraints are nearly degenerate. We study the implementation, convergence properties and performance of an algorithm that employs this idea. The implementation considerations include the choice and automatic adjustment of the tolerance that defines the small residuals, the calculation of the search directions, and the updating of second derivative approximations. The main convergence theorem imposes no conditions on the constraints except for boundedness of the feasible region. The numerical results indicate that a Fortran implementation of our algorithm is much more reliable than the software that was tested by Hock and Schittkowski (1981). Therefore the algorithm seems to be very suitable for general use, and it is particularly appropriate for semi-infinite programming calculations that have many linear constraints that come from discretizations of continua.     

A Computational Study of the Homogeneous Algorithm for Large-scale Convex Optimization

Recently the authors have proposed a homogeneous and self-dual algorithm for solving the monotone complementarity problem (MCP) [5]. The algorithm is a single phase interior-point type method; nevertheless, it yields either an approximate optimal solution or detects a possible infeasibility of the problem. In this paper we specialize the algorithm to the solution of general smooth convex optimization problems, which also possess nonlinear inequality constraints and free variables. We discuss an implementation of the algorithm for large-scale sparse convex optimization. Moreover, we present computational results for solving quadratically constrained quadratic programming and geometric programming problems, where some of the problems contain more than 100,000 constraints and variables. The results indicate that the proposed algorithm is also practically efficient.     

Progressive genetic algorithm for solution of optimization problems with nonlinear equality and inequality constraints

A new approach, identified as progressive genetic algorithm (PGA), is proposed for the solutions of optimization problems with nonlinear equality and inequality constraints. Based on genetic algorithms (GAs) and iteration method, PGA divides the optimization process into two steps; iteration and search steps. In the iteration step, the constraints of the original problem are linearized using truncated Taylor series expansion, yielding an approximate problem with linearized constraints. In the search step, GA is applied to the problem with linearized constraints for the local optimal solution. The final solution is obtained from a progressive iterative process. Application of the proposed method to two simple examples is given to demonstrate the algorithm.     

An improved penalty function method for solving constrained parameter optimization problems

An effective algorithm is described for solving the general constrained parameter optimization problem. The method is quasi-second-order and requires only function and gradient information. An exterior point penalty function method is used to transform the constrained problem into a sequence of unconstrained problems. The penalty weightr is chosen as a function of the pointx such that the sequence of optimization problems is computationally easy. A rank-one optimization algorithm is developed that takes advantage of the special properties of the augmented performance index. The optimization algorithm accounts for the usual difficulties associated with discontinuous second derivatives of the augmented index. Finite convergence is exhibited for a quadratic performance index with linear constraints; accelerated convergence is demonstrated for nonquadratic indices and nonlinear constraints. A computer program has been written to implement the algorithm and its performance is illustrated in fourteen test problems.     

A GENERALIZED PROJECTION-SUCCESSIVE LINEAREQUATIONS ALGORITHM FOR NONLINEARLY EQUALITy AND INEQUALITY CONSTRAINED OPTIMIZATION AND ITS RATE OF CONVERGENCE

In this paper, a new superlinearly convergent algorithm is presented for optimization problems with general nonlineer equality and inequality Constraints, Comparing with other methods for these problems, the algorithm has two main advantages. First, it doesn‘t solve anyquadratic programming (QP), and its search directions are determined by the generalized projection technique and the solutions of two systems of linear equations. Second, the sequential points generated by the algoritbh satisfy all inequity constraints and its step-length is computed by the straight line search,The algorithm is proved to possesa global and auperlinear convergence.     

Sequential conjugate-gradient-restoration algorithm for optimal control problems. Part 2. Examples

In Ref. 1, Heideman and Levy developed the sequential conjugate-gradient-restoration algorithm for minimizing a functional subject to differential constraints and terminal constraints. In this report, several numerical examples are presented, some pertaining to a quadratic functional subject to linear constraints and some pertaining to a nonquadratic functional subject to nonlinear constraints. These examples demonstrate the feasibility as well as the rapid convergence characteristic of the sequential conjugate-gradient-restoration algorithm.     

A feasible direction method for the semidefinite program with box constraints

In this paper, we try to solve the semidefinite program with box constraints. Since the traditional projection method for constrained optimization with box constraints is not suitable to the semidefinite constraints, we present a new algorithm based on the feasible direction method. In the paper, we discuss two cases: the objective function in semidefinite programming is linear and nonlinear, respectively. We establish the convergence of our algorithm, and report the numerical experiments which show the effectiveness of the algorithm.     

An optimization-based heuristic for the robotic cell problem

This study investigates an optimization-based heuristic for the robotic cell problem. This problem arises in automated cells and is a complex flow shop problem with a single transportation robot and a blocking constraint. We propose an approximate decomposition algorithm. The proposed approach breaks the problem into two scheduling problems that are solved sequentially: a flow shop problem with additional constraints (blocking and transportation times) and a single machine problem with precedence constraints, time lags, and setup times. For each of these problems, we propose an exact branch-and-bound algorithm. Also, we describe a genetic algorithm that includes, as a mutation operator, a local search procedure. We report the results of a computational study that provides evidence that the proposed optimization-based approach delivers high-quality solutions and consistently outperforms the genetic algorithm. However, the genetic algorithm delivers reasonably good solutions while requiring significantly shorter CPU times.     

An exact algorithm for the precedence-constrained single-machine scheduling problem

This study proposes an efficient exact algorithm for the precedence-constrained single-machine scheduling problem to minimize total job completion cost where machine idle time is forbidden. The proposed algorithm is based on the SSDP (Successive Sublimation Dynamic Programming) method and is an extension of the authors’ previous algorithms for the problem without precedence constraints. In this method, a lower bound is computed by solving a Lagrangian relaxation of the original problem via dynamic programming and then it is improved successively by adding constraints to the relaxation until the gap between the lower and upper bounds vanishes. Numerical experiments will show that the algorithm can solve all instances with up to 50 jobs of the precedence-constrained total weighted tardiness and total weighted earliness–tardiness problems, and most instances with 100 jobs of the former problem.     

An algorithm for solving the problem of convex programming with several objective functions

This work aims to establish an algorithm for solving the problem of convex programming with several objective-functions, with linear constraints. Starting from the idea of Rosen’s algorithm for solving the problem of convex programming with linear constraints, and taking into account the solution concept from multidimensional programming, represented by a program which reaches ”the best compromise”, we are extending this method in the case of multidimensional programming. The concept of direction of minimization is introduced, and a necessary and sufficient condition is given for as ∈R ⁿ direction to be a direction of minimization, according to the values of a criteria ensemble in a given point. The algorithm is interactive, and the intervention of the decident is minimal. The two numerical examples presented at the end validate the algorithm.     

Greedy primal-dual algorithm for dynamic resource allocation in complex networks

In Stolyar (Queueing Systems 50 (2005) 401–457) a dynamic control strategy, called greedy primal-dual (GPD) algorithm, was introduced for the problem of maximizing queueing network utility subject to stability of the queues, and was proved to be (asymptotically) optimal. (The network utility is a concave function of the average rates at which the network generates several “commodities.”) Underlying the control problem of Stolyar (Queueing Systems 50 (2005) 401–457) is a convex optimization problem subject to a set of linear constraints. In this paper we introduce a generalized GPD algorithm, which applies to the network control problem with additional convex (possibly non-linear) constraints on the average commodity rates. The underlying optimization problem in this case is a convex problem subject to convex constraints. We prove asymptotic optimality of the generalized GPD algorithm. We illustrate key features and applications of the algorithm on simple examples. AMS Subject Classifications: 90B15 · 90C25 · 60K25 · 68M12