Bee colony optimization for scheduling independent tasks to identical processors

The static scheduling of independent tasks on homogeneous multiprocessor systems is studied in this paper. This problem is treated by the Bee Colony Optimization (BCO) meta-heuristic. The BCO algorithm belongs to the class of stochastic swarm optimization methods inspired by the foraging habits of bees in nature. To investigate the performance of the proposed method extensive numerical experiments are performed. Our BCO algorithm is able to obtain the optimal value of the objective function in the majority of test examples known from literature. The deviation of non-optimal solutions from the optimal ones in our test examples is at most 2%. The CPU times required to find the best solutions by BCO are significantly smaller than the corresponding times required by the CPLEX optimization solver. Moreover, our BCO is competitive with state-of-the-art methods for similar problems, with respect to both solution quality and running time. The stability of BCO is examined through multiple executions and it is shown that solution deviation is less than 1%.     

A Stochastic/Perturbation Global Optimization Algorithm for Distance Geometry Problems

We present a new global optimization approach for solving exactly or inexactly constrained distance geometry problems. Distance geometry problems are concerned with determining spatial structures from measurements of internal distances. They arise in the structural interpretation of nuclear magnetic resonance data and in the prediction of protein structure. These problems can be naturally formulated as global optimization problems which generally are large and difficult. The global optimization method that we present is related to our previous stochastic/perturbation global optimization methods for finding minimum energy configurations, but has several key differences that are important to its success. Our computational results show that the method readily solves a set of artificial problems introduced by Moré and Wu that have up to 343 atoms. On a set of considerably more difficult protein fragment problems introduced by Hendrickson, the method solves all the problems with up to 377 atoms exactly, and finds nearly exact solution for all the remaining problems which have up to 777 atoms. These preliminary results indicate that this approach has very good promise for helping to solve distance geometry problems.     

The hierarchical and social decision making procedure for bi-objective optimization problems by using the simple majority decision rule

Multi-objective optimization has been successfully applied to problems of industrial design, problems of quality control and production management, and problems of finance. The theme of these applications is how to choose the best solution for the decision makers out of a set of non-inferior solutions to a multi-objective optimization problem. For this purpose, an optimization model with hierarchical structure, whose lower problem is a multi-objective optimization problem and the upper problem is a preference optimization problem on a set of non-inferior solutions, must be constructed. This kind of hierarchical problems have been previously analyzed only with regard to linear programming problems by Benson[6]. In this paper, an algorithm is derived that provides a solution as a social choice, obtained by aggregating plural decision-makers' preferences. In the case of the simple majority rule, the bi-objective problem is transformed into an -parameter choice problem, and the golden section method is applied. The availability of the approach is demonstrated with the means of an illustrative example.Technische Universität BerlinFaculty of Science and Technology, Keio University     

Barrier subgradient method

In this paper we develop a new affine-invariant primal–dual subgradient method for nonsmooth convex optimization problems. This scheme is based on a self-concordant barrier for the basic feasible set. It is suitable for finding approximate solutions with certain relative accuracy. We discuss some applications of this technique including fractional covering problem, maximal concurrent flow problem, semidefinite relaxations and nonlinear online optimization. For all these problems, the rate of convergence of our method does not depend on the problem’s data.     

An improved hybrid global optimization method for protein tertiary structure prediction

First principles approaches to the protein structure prediction problem must search through an enormous conformational space to identify low-energy, near-native structures. In this paper, we describe the formulation of the tertiary structure prediction problem as a nonlinear constrained minimization problem, where the goal is to minimize the energy of a protein conformation subject to constraints on torsion angles and interatomic distances. The core of the proposed algorithm is a hybrid global optimization method that combines the benefits of the αBB deterministic global optimization approach with conformational space annealing. These global optimization techniques employ a local minimization strategy that combines torsion angle dynamics and rotamer optimization to identify and improve the selection of initial conformations and then applies a sequential quadratic programming approach to further minimize the energy of the protein conformations subject to constraints. The proposed algorithm demonstrates the ability to identify both lower energy protein structures, as well as larger ensembles of low-energy conformations.     

A global optimization method for the molecular replacement problem in X-ray crystallography

The primary technique for determining the three-dimensional structure of a protein molecule is X-ray crystallography, from which the molecular replacement (MR) problem often arises as a critical step. The MR problem is a global optimization problem to locate an optimal position of a model protein so that at this position the model will produce calculated intensities closest to those observed from an X-ray crystallography experiment involving a protein with unknown but similar atomic structure. Improving the applicability and robustness of MR methods is an important research topic because commonly used traditional MR methods, though often successful, have their limitations in solving difficult problems.We introduce a new global optimization strategy that combines a coarse-grid search, using a surrogate function, with extensive multi-start local optimization. A new MR code, called SOMoRe, based on this strategy is developed and tested on four realistic problems, including two difficult problems that traditional MR codes failed to solve directly. SOMoRe was able to solve each test problem without any complication, and SOMoRe solved an MR problem using a less complete model than the models required by three other programs. These results indicate that the new method is promising and should enhance the applicability and robustness of the MR methodology.     

Maximax and minimax rearrangement optimization problems

In this paper we present a general theory concerning two rearrangement optimization problems; one of maximization and the other of minimization type. The structure of the cost functional allows to formulate the two problems as maximax and minimax optimization problems. The latter proves to be far more interesting than the former. As an application of the theory we investigate a shape optimization problem which has already been addressed by other authors; however, here we prove our method is more efficient, and has the advantage that it captures more features of the optimal solutions than those obtained by others. The paper ends with a special case of the minimax problem, where we are able to obtain a minimum size estimate related to the optimal solution.     

Heuristic Methods for Large Centroid Clustering Problems

This article presents new heuristic methods for solving a class of hard centroid clustering problems including the p-median, the sum-of-squares clustering and the multi-source Weber problems. Centroid clustering is to partition a set of entities into a given number of subsets and to find the location of a centre for each subset in such a way that a dissimilarity measure between the entities and the centres is minimized. The first method proposed is a candidate list search that produces good solutions in a short amount of time if the number of centres in the problem is not too large. The second method is a general local optimization approach that finds very good solutions. The third method is designed for problems with a large number of centres; it decomposes the problem into subproblems that are solved independently. Numerical results show that these methods are efficient—dozens of best solutions known to problem instances of the literature have been improved—and fast, handling problem instances with more than 85,000 entities and 15,000 centres—much larger than those solved in the literature. The expected complexity of these new procedures is discussed and shown to be comparable to that of an existing method which is known to be very fast.     

An algorithm for nonlinear optimization problems with binary variables

One of the challenging optimization problems is determining the minimizer of a nonlinear programming problem that has binary variables. A vexing difficulty is the rate the work to solve such problems increases as the number of discrete variables increases. Any such problem with bounded discrete variables, especially binary variables, may be transformed to that of finding a global optimum of a problem in continuous variables. However, the transformed problems usually have astronomically large numbers of local minimizers, making them harder to solve than typical global optimization problems. Despite this apparent disadvantage, we show that the approach is not futile if we use smoothing techniques. The method we advocate first convexifies the problem and then solves a sequence of subproblems, whose solutions form a trajectory that leads to the solution. To illustrate how well the algorithm performs we show the computational results of applying it to problems taken from the literature and new test problems with known optimal solutions.     

GASUB: finding global optima to discrete location problems by a genetic-like algorithm

In many discrete location problems, a given number s of facility locations must be selected from a set of m potential locations, so as to optimize a predetermined fitness function. Most of such problems can be formulated as integer linear optimization problems, but the standard optimizers only are able to find one global optimum. We propose a new genetic-like algorithm, GASUB, which is able to find a predetermined number of global optima, if they exist, for a variety of discrete location problems. In this paper, a performance evaluation of GASUB in terms of its effectiveness (for finding optimal solutions) and efficiency (computational cost) is carried out. GASUB is also compared to MSH, a multi-start substitution method widely used for location problems. Computational experiments with three types of discrete location problems show that GASUB obtains better solutions than MSH. Furthermore, the proposed algorithm finds global optima in all tested problems, which is shown by solving those problems by Xpress-MP, an integer linear programing optimizer (21). Results from testing GASUB with a set of known test problems are also provided.     

Optimizing simulated annealing schedules with genetic programming

Combinatorial optimization problems are encountered in many areas of science and engineering. Most of these problems are too difficult to be solved optimally, and hence heuristics are used to obtain “good” solutions in reasonable time. One heuristic that has been successfully applied to a variety of problems is Simulated Annealing. The performance of Simulated Annealing depends on the appropriate choice of a key parameter, the annealing schedule. Researchers usually experiment with some manually created annealing schedules and then use the one that performs best in their algorithms.This work replaces this manual search by Genetic Programming, a method based on natural evolution. We demonstrate the potential of this new approach by optimizing the annealing schedule for a well-known combinatorial optimization problem, the Quadratic Assignment Problem. We introduce two new algorithms for solving the Quadratic Assignment Problem that perform extremely well and outperform existing Simulated Annealing algorithms.     

The Exactness Property of the Vector Exact l1 Penalty Function Method in Nondifferentiable Invex Multiobjective Programming

In this article, the vector exact l₁ penalty function method used for solving nonconvex nondifferentiable multiobjective programming problems is analyzed. In this method, the vector penalized optimization problem with the vector exact l₁ penalty function is defined. Conditions are given guaranteeing the equivalence of the sets of (weak) Pareto optimal solutions of the considered nondifferentiable multiobjective programming problem and of the associated vector penalized optimization problem with the vector exact l₁ penalty function. This equivalence is established for nondifferentiable invex vector optimization problems. Some examples of vector optimization problems are presented to illustrate the results established in the article.     

Reinforced Angle-Based Multicategory Support Vector Machines

The support vector machine (SVM) is a very popular classification tool with many successful applications. It was originally designed for binary problems with desirable theoretical properties. Although there exist various multicategory SVM (MSVM) extensions in the literature, some challenges remain. In particular, most existing MSVMs make use of k classification functions for a k-class problem, and the corresponding optimization problems are typically handled by existing quadratic programming solvers. In this article, we propose a new group of MSVMs, namely, the reinforced angle-based MSVMs (RAMSVMs), using an angle-based prediction rule with k ? 1 functions directly. We prove that RAMSVMs can enjoy Fisher consistency. Moreover, we show that the RAMSVM can be implemented using the very efficient coordinate descent algorithm on its dual problem. Numerical experiments demonstrate that our method is highly competitive in terms of computational speed, as well as classification prediction performance. Supplemental materials for the article are available online.     

Optimal <Emphasis Type="Italic">L</Emphasis><Superscript>1</Superscript>-Control in Coefficients for Dirichlet Elliptic Problems: <Emphasis Type="Italic">W</Emphasis>-Optimal Solutions

In this paper, we study a Dirichlet optimal control problem associated with a linear elliptic equation the coefficients of which we take as controls in the class of integrable functions. The coefficients may degenerate and, therefore, the problems may exhibit the so-called Lavrentieff phenomenon and non-uniqueness of weak solutions. We consider the solvability of this problem in the class of W-variational solutions. Using a concept of variational convergence of constrained minimization problems in variable spaces, we prove the existence of W-solutions to the optimal control problem and provide the way for their approximation. We emphasize that control problems of this type are important in material and topology optimization as well as in damage or life-cycle optimization.     

A Population-based Approach for Hard Global Optimization Problems based on Dissimilarity Measures

When dealing with extremely hard global optimization problems, i.e. problems with a large number of variables and a huge number of local optima, heuristic procedures are the only possible choice. In this situation, lacking any possibility of guaranteeing global optimality for most problem instances, it is quite difficult to establish rules for discriminating among different algorithms. We think that in order to judge the quality of new global optimization methods, different criteria might be adopted like, e.g.:

Simulated annealing with asymptotic convergence for nonlinear constrained optimization

In this paper, we present constrained simulated annealing (CSA), an algorithm that extends conventional simulated annealing to look for constrained local minima of nonlinear constrained optimization problems. The algorithm is based on the theory of extended saddle points (ESPs) that shows the one-to-one correspondence between a constrained local minimum and an ESP of the corresponding penalty function. CSA finds ESPs by systematically controlling probabilistic descents in the problem-variable subspace of the penalty function and probabilistic ascents in the penalty subspace. Based on the decomposition of the necessary and sufficient ESP condition into multiple necessary conditions, we present constraint-partitioned simulated annealing (CPSA) that exploits the locality of constraints in nonlinear optimization problems. CPSA leads to much lower complexity as compared to that of CSA by partitioning the constraints of a problem into significantly simpler subproblems, solving each independently, and resolving those violated global constraints across the subproblems. We prove that both CSA and CPSA asymptotically converge to a constrained global minimum with probability one in discrete optimization problems. The result extends conventional simulated annealing (SA), which guarantees asymptotic convergence in discrete unconstrained optimization, to that in discrete constrained optimization. Moreover, it establishes the condition under which optimal solutions can be found in constraint-partitioned nonlinear optimization problems. Finally, we evaluate CSA and CPSA by applying them to solve some continuous constrained optimization benchmarks and compare their performance to that of other penalty methods.     

Large-scale linear programming techniques for the design of protein folding potentials

We present large-scale optimization techniques to model the energy function that underlies the folding process of proteins. Linear Programming is used to identify parameters in the energy function model, the objective being that the model predict the structure of known proteins correctly. Such trained functions can then be used either for ab-initio prediction or for recognition of unknown structures. In order to obtain good energy models we need to be able to solve dense Linear Programming Problems with tens (possibly hundreds) of millions of constraints in a few hundred parameters, which we achieve by tailoring and parallelizing the interior-point code PCx.     

Reference variable methods of solving min–max optimization problems

In this paper, reference variable methods are proposed for solving nonlinear Minmax optimization problems with unconstraint or constraints for the first time, it uses reference decision vectors to improve the methods in Vincent and Goh (J Optim Theory Appl 75:501–519, 1992) such that its algorithm is convergent. In addition, a new method based on KKT conditions of min or max constrained optimization problems is also given for solving the constrained minmax optimization problems, it makes the constrained minmax optimization problems a problem of solving nonlinear equations by a complementarily function. For getting all minmax optimization solutions, the cost function f(x, y) can be constrained as M ₁ < f(x, y) < M ₂ by using different real numbers M ₁ and M ₂. To show effectiveness of the proposed methods, some examples are taken to compare with results in the literature, and it is easy to find that the proposed methods can get all minmax optimization solutions of minmax problems with constraints by using different M ₁ and M ₂, this implies that the proposed methods has superiority over the methods in the literature (that is based on different initial values to get other minmax optimization solutions).     

Multi-funnel optimization using Gaussian underestimation

In several applications, underestimation of functions has proven to be a helpful tool for global optimization. In protein–ligand docking problems as well as in protein structure prediction, single convex quadratic underestimators have been used to approximate the location of the global minimum point. While this approach has been successful for basin-shaped functions, it is not suitable for energy functions with more than one distinct local minimum with a large magnitude. Such functions may contain several basin-shaped components and, thus, cannot be underfitted by a single convex underestimator. In this paper, we propose using an underestimator composed of several negative Gaussian functions. Such an underestimator can be computed by solving a nonlinear programming problem, which minimizes the error between the data points and the underestimator in the L ¹ norm. Numerical results for simulated and actual docking energy functions are presented.     

Finding preferred subsets of Pareto optimal solutions

Multi-objective optimization algorithms can generate large sets of Pareto optimal (non-dominated) solutions. Identifying the best solutions across a very large number of Pareto optimal solutions can be a challenge. Therefore it is useful for the decision-maker to be able to obtain a small set of preferred Pareto optimal solutions. This paper analyzes a discrete optimization problem introduced to obtain optimal subsets of solutions from large sets of Pareto optimal solutions. This discrete optimization problem is proven to be NP-hard. Two exact algorithms and five heuristics are presented to address this problem. Five test problems are used to compare the performances of these algorithms and heuristics. The results suggest that preferred subset of Pareto optimal solutions can be efficiently obtained using the heuristics, while for smaller problems, exact algorithms can be applied.