Global Optimization Performance Measures for Generalized Hill Climbing Algorithms

Generalized hill climbing algorithms provide a framework for modeling several local search algorithms for hard discrete optimization problems. This paper introduces and analyzes generalized hill climbing algorithm performance measures that reflect how effectively an algorithm has performed to date in visiting a global optimum and how effectively an algorithm may pes]rform in the future in visiting such a solution. These measures are also used to obtain a necessary asymptotic convergence (in probability) condition to a global optimum, which is then used to show that a common formulation of threshold accepting does not converge. These measures assume particularly simple forms when applied to specific search strategies such as Monte Carlo search and threshold accepting.     

A new branch and bound algorithm for solving quadratic programs with linear complementarity constraints

In order to find a global solution for a quadratic program with linear complementarity constraints (QPLCC) more quickly than some existing methods, we consider to embed a local search method into a global search method. To say more specifically, in a branch-and-bound algorithm for solving QPLCC, when we find a new feasible solution to the problem, we utilize an extreme point algorithm to obtain a locally optimal solution which can provide a better bound and help us to trim more branches. So, the global algorithm can be accelerated. A preliminary numerical experiment was conducted which supports the new algorithm.     

Bound constrained interval global optimization in the COCONUT Environment

We introduce a new interval global optimization method for solving bound constrained problems. The method originates from a small standalone software and is implemented in the COCONUT Environment, a framework designed for the development of complex algorithms, containing numerous state-of-the-art methods in a common software platform. The original algorithm is enhanced by various new methods implemented in COCONUT, regarding both interval function evaluations (such as first and second order derivatives with backward automatic differentiation, slopes, slopes of derivatives, bicentered forms, evaluations on the Karush–John conditions, etc.) and algorithmic elements (inclusion/exclusion boxes, local search, constraint propagation). This resulted in a substantial performance increase as compared to the original code. During the selection of the best combination of options, we performed comparison tests that gave empirical answers to long-lasting algorithmic questions (such as whether to use interval gradients or use slopes instead), that have never been studied numerically in such detail before. The new algorithm, called coco_gop_ex, was tested against the prestigious BARON software on an extensive set of bound constrained problems. We found that in addition to accepting a wider class of bound constrained problems and providing more output information (by locating all global minimizers), coco_gop_ex is competitive with BARON in terms of the solution success rates (with the exception of a set of nonlinear least squares problems), and it often outperforms BARON in running time. In particular, coco_gop_ex was around 21 % faster on average over the set of problems solved by both software systems.     

Continuous functions minimization by dynamic random search technique

Random search technique is the simplest one of the heuristic algorithms. It is stated in the literature that the probability of finding global minimum is equal to 1 by using the basic random search technique, but it takes too much time to reach the global minimum. Improving the basic random search technique may decrease the solution time. In this study, in order to obtain the global minimum fastly, a new random search algorithm is suggested. This algorithm is called as the Dynamic Random Search Technique (DRASET). DRASET consists of two phases, which are general search and local search based on general solution. Knowledge related to the best solution found in the process of general search is kept and then that knowledge is used as initial value of local search. DRASET’s performance was experimented with 15 test problems and satisfactory results were obtained.     

A particle swarm optimization-based hybrid algorithm for minimum concave cost network flow problems

Traditionally, minimum cost transshipment problems have been simplified as linear cost problems, which are not practical in real applications. Some advanced local search algorithms have been developed to solve concave cost bipartite network problems. These have been found to be more effective than the traditional linear approximation methods and local search methods. Recently, a genetic algorithm and an ant colony system algorithm were employed to develop two global search algorithms for solving concave cost transshipment problems. These two global search algorithms were found to be more effective than the advanced local search algorithms for solving concave cost transshipment problems. Although the particle swarm optimization algorithm has been used to obtain good results in many applications, to the best of our knowledge, it has not yet been applied in minimum concave cost network flow problems. Thus, in this study, we employ an arc-based particle swarm optimization algorithm, coupled with some genetic algorithm and threshold accepting method techniques, as well as concave cost network heuristics, to develop a hybrid global search algorithm for efficiently solving minimum cost network flow problems with concave arc costs. The proposed algorithm is evaluated by solving several randomly generated network flow problems. The results indicate that the proposed algorithm is more effective than several other recently designed methods, such as local search algorithms, genetic algorithms and ant colony system algorithms, for solving minimum cost network flow problems with concave arc costs.     

Interval Branch and Bound with Local Sampling for Constrained Global Optimization

In this article, we introduce a global optimization algorithm that integrates the basic idea of interval branch and bound, and new local sampling strategies along with an efficient data structure. Also included in the algorithm are procedures that handle constraints. The algorithm is shown to be able to find all the global optimal solutions under mild conditions. It can be used to solve various optimization problems. The local sampling (even if done stochastically) is used only to speed up the convergence and does not affect the fact that a complete search is done. Results on several examples of various dimensions ranging from 1 to 100 are also presented to illustrate numerical performance of the algorithm along with comparison with another interval method without the new local sampling and several noninterval methods. The new algorithm is seen as the best performer among those tested for solving multi-dimensional problems.     

Parallel Information Algorithm with Local Tuning for Solving Multidimensional GO Problems

In this paper we propose a new parallel algorithm for solving global optimization (GO) multidimensional problems. The method unifies two powerful approaches for accelerating the search: parallel computations and local tuning on the behavior of the objective function. We establish convergence conditions for the algorithm and theoretically show that the usage of local information during the global search permits to accelerate solving the problem significantly. Results of numerical experiments executed with 100 test functions are also reported.     

Optimal control of bioprocess systems using hybrid numerical optimization algorithms

In the paper, we consider the bioprocess system optimal control problem. Generally speaking, it is very difficult to solve this problem analytically. To obtain the numerical solution, the problem is transformed into a parameter optimization problem with some variable bounds, which can be efficiently solved using any conventional optimization algorithms, e.g. the improved Broyden–Fletcher–Goldfarb–Shanno algorithm. However, in spite of the improved Broyden–Fletcher–Goldfarb–Shanno algorithm is very efficient for local search, the solution obtained is usually a local extremum for non-convex optimal control problems. In order to escape from the local extremum, we develop a novel stochastic search method. By performing a large amount of numerical experiments, we find that the novel stochastic search method is excellent in exploration, while bad in exploitation. In order to improve the exploitation, we propose a hybrid numerical optimization algorithm to solve the problem based on the novel stochastic search method and the improved Broyden–Fletcher–Goldfarb–Shanno algorithm. Convergence results indicate that any global optimal solution of the approximate problem is also a global optimal solution of the original problem. Finally, two bioprocess system optimal control problems illustrate that the hybrid numerical optimization algorithm proposed by us is low time-consuming and obtains a better cost function value than the existing approaches.     

Global search algorithms for minimum concave-cost network flow problems

We present algorithms for the single-source uncapacitated version of the minimum concave cost network flow problem. Each algorithm exploits the fact that an extreme feasible solution corresponds to a sub-tree of the original network. A global search heuristic based on random extreme feasible initial solutions and local search is developed. The algorithm is used to evaluate the complexity of the randomly generated test problems. An exact global search algorithm is developed, based on enumerative search of rooted subtrees. This exact technique is extended to bound the search based on cost properties and linear underestimation. The technique is accelerated by exploiting the network structure.     

A Theoretical and Numerical Comparison of Some Semismooth Algorithms for Complementarity Problems

In this paper we introduce a general line search scheme which easily allows us to define and analyze known and new semismooth algorithms for the solution of nonlinear complementarity problems. We enucleate the basic assumptions that a search direction to be used in the general scheme has to enjoy in order to guarantee global convergence, local superlinear/quadratic convergence or finite convergence. We examine in detail several different semismooth algorithms and compare their theoretical features and their practical behavior on a set of large-scale problems.