Constrained Global Optimization Research Articles

A Filled Function Approach for Nonsmooth Constrained Global Optimization

A novel filled function is given in this paper to find a global minima for a nonsmooth constrained optimization problem. First, a modified concept of the filled function for nonsmooth constrained global optimization is introduced, and a filled function, which makes use of the idea of the filled function for unconstrained optimization and penalty function for constrained optimization, is proposed. Then, a solution algorithm based on the proposed filled function is developed. At last, some preliminary numerical results are reported. The results show that the proposed approach is promising.

A simulated annealing driven multi-start algorithm for bound constrained global optimization

A derivative-free simulated annealing driven multi-start algorithm for continuous global optimization is presented. We first propose a trial point generation scheme in continuous simulated annealing which eliminates the need for the gradient-based trial point generation. We then suitably embed the multi-start procedure within the simulated annealing algorithm. We modify the derivative-free pattern search method and use it as the local search in the multi-start procedure. We study the convergence properties of the algorithm and test its performance on a set of 50 problems. Numerical results are presented which show the robustness of the algorithm. Numerical comparisons with a gradient-based simulated annealing algorithm and three population-based global optimization algorithms show that the new algorithm could offer a reasonable alternative to many currently available global optimization algorithms, specially for problems requiring ‘direct search’ type algorithm.

Constrained global optimization of multivariate polynomials using Bernstein branch and prune algorithm

We propose an algorithm for constrained global optimization to tackle non-convex nonlinear multivariate polynomial programming problems. The proposed Bernstein branch and prune algorithm is based on the Bernstein polynomial approach. We introduce several new features in this proposed algorithm to make the algorithm more efficient. We first present the Bernstein box consistency and Bernstein hull consistency algorithms to prune the search regions. We then give Bernstein contraction algorithm to avoid the computation of Bernstein coefficients after the pruning operation. We also include a new Bernstein cut-off test based on the vertex property of the Bernstein coefficients. The performance of the proposed algorithm is numerically tested on 13 benchmark problems. The results of the tests show the proposed algorithm to be overall considerably superior to existing method in terms of the chosen performance metrics.

INTLAB implementation of an interval global optimization algorithm

We describe a new implementation of an interval optimization algorithm with focus on the software related issues. The algorithm implemented in MATLAB that uses the INTLAB package supporting interval calculations and automatic differentiation solves the bound constrained global optimization problem. The method itself is a simplified version of those interval techniques much investigated in the past, that was first developed from the global optimization algorithm of the Numerical Toolbox for Verified Computing. According to the numerical studies completed, the new INTLAB-based implementation is closely as efficient as its C-XSC-based basis algorithm – with the exception of the CPU time needed (the longer computations are due to the interpreter nature of MATLAB).

A one-parameter filled function method applied to nonsmooth constrained global optimization

In this paper, we propose a filled function method for solving nonsmooth constrained global optimization problems. Based on a new definition of the filled function, a more practical one-parameter filled function is constructed which overcomes some drawbacks of the previous filled functions. Then a corresponding algorithm is presented. It attains a local minimizer by implementing a local search procedure, and finds a better local minimizer gradually by optimizing the filled function constructed on the minimizer, previously found. By repeating these steps, a global minimizer is obtained. Numerical experiments are presented to show the practicability of the proposed filled function method. In the end, extension conceivable applications are given in order to evaluate the merits of this method.

A filled function method applied to nonsmooth constrained global optimization

The filled function method is an effective approach to find a global minimizer. In this paper, based on a new definition of the filled function for nonsmooth constrained programming problems, a one-parameter filled function is constructed to improve the efficiency of numerical computation. Then a corresponding algorithm is presented. It is a global optimization method which modify the objective function as a filled function, and which find a better local minimizer gradually by optimizing the filled function constructed on the minimizer previously found. Illustrative examples are provided to demonstrate the efficiency and reliability of the proposed filled function method.

Extensions of a Multistart Clustering Algorithm for Constrained Global Optimization Problems

Here, we consider the solution of constrained global optimization problems, such as those arising from the fields of chemical and biosystems engineering. These problems are frequently formulated (o...

Solutions and optimality criteria for nonconvex constrained global optimization problems with connections between canonical and Lagrangian duality

This paper presents a canonical duality theory for solving a general nonconvex quadratic minimization problem with nonconvex constraints. By using the canonical dual transformation developed by the first author, the nonconvex primal problem can be converted into a canonical dual problem with zero duality gap. A general analytical solution form is obtained. Both global and local extrema of the nonconvex problem can be identified by the triality theory associated with the canonical duality theory. Illustrative applications to quadratic minimization with multiple quadratic constraints, box/integer constraints, and general nonconvex polynomial constraints are discussed, along with insightful connections to classical Lagrangian duality. Criteria for the existence and uniqueness of optimal solutions are presented. Several numerical examples are provided.

Solving nonlinearly constrained global optimization problem via an auxiliary function method

This paper considers the nonlinearly constrained continuous global minimization problem. Based on the idea of the penalty function method, an auxiliary function, which has approximately the same global minimizers as the original problem, is constructed. An algorithm is developed to minimize the auxiliary function to find an approximate constrained global minimizer of the constrained global minimization problem. The algorithm can escape from the previously converged local minimizers, and can converge to an approximate global minimizer of the problem asymptotically with probability one. Numerical experiments show that it is better than some other well known recent methods for constrained global minimization problems.

Digital Pattern Search and Its Hybridization with Genetic Algorithms for Bound Constrained Global Optimization

In this paper, we present a recently developed pattern search method called Genetic Pattern Search algorithm (GPSA) for the global optimization of cost function subject to simple bounds. GPSA is a combined global optimization method using genetic algorithm (GA) and Digital Pattern Search (DPS) method, which has the digital structure represented by binary strings and guarantees convergence to stationary points from arbitrary starting points. The performance of GPSA is validated through extensive numerical experiments on a number of well known functions and on robot walking application. The optimization results confirm that GPSA is a robust and efficient global optimization method.

Interval algorithm for global numerical optimization

The interval optimization algorithm shows great advantages in bound constrained global optimization. An interval algorithm is presented in this article based on a new selection criterion. The selection criterion is proposed based on numerical experiments and the parameter pf* designed by Casado, Garcia and Csendes in 2000. The proposed criterion at each iteration selects some intervals of which the number is not greater than a constant so that the possible memory problem during the implementation of the algorithm is avoided and the running time of the algorithm is decreased, when the dimension of the problem is increasing. Based on the selection criterion, the proposed algorithm is implemented for a wide set of tested functions which includes easy and hard problems. Numerical experiments show that the proposed algorithm is efficient.

Estimation of physically realizable Mueller matrices from experiments using global constrained optimization

One can explicitly retrieve physically realizable Mueller matrices from quantified intensity data even in the presence of noise. This is done by integrating the physical realizability criterion obtained by Givens and Kostinski, [J. Mod. Opt. 40, 471 (1993)], as an active constraint in a global optimization process. Among different global optimization techniques, two of them have been tested and their robustness analyzed: a deterministic approach based on sequential quadratic programming and a stochastic approach based on constrained simulated annealing algorithms are implemented for this purpose. We illustrate the validity of both methods on experimental data and on the inadmissible Mueller matrix given by Howell, [Appl. Opt. 18, No. 6, 808-812 (1979)]. In comparison, the constrained simulated annealing method produced higher accuracy with similar computing time.

On the Convergence of Augmented Lagrangian Methods for Constrained Global Optimization

In this paper, we present new convergence properties of the primal-dual method based on four types of augmented Lagrangian functions in the context of constrained global optimization. Convergence to a global optimal solution is first established for a basic primal-dual scheme under standard conditions. We then prove this convergence property for a modified augmented Lagrangian method using a safeguarding strategy without appealing to the boundedness assumption of the multiplier sequence. We further show that, under the same weaker conditions, the convergence to a global optimal solution can still be achieved by either modifying the multiplier updating rule or normalizing the multipliers in augmented Lagrangian methods.

The GLOBAL optimization method revisited

The multistart clustering global optimization method called GLOBAL has been introduced in the 1980s for bound constrained global optimization problems with black-box type objective function. Since then the technological environment has been changed much. The present paper describes shortly the revisions and updates made on the involved algorithms to utilize the novel technologies, and to improve its reliability. We discuss in detail the results of the numerical comparison with the old version and with C-GRASP, a continuous version of the GRASP method. According to these findings, the new version of GLOBAL is both more reliable and more efficient than the old one, and it compares favorably with C-GRASP too.

Parallel radial basis function methods for the global optimization of expensive functions

We introduce a master–worker framework for parallel global optimization of computationally expensive functions using response surface models. In particular, we parallelize two radial basis function (RBF) methods for global optimization, namely, the RBF method by Gutmann [Gutmann, H.M., 2001a. A radial basis function method for global optimization. Journal of Global Optimization 19(3), 201–227] (Gutmann-RBF) and the RBF method by Regis and Shoemaker [Regis, R.G., Shoemaker, C.A., 2005. Constrained global optimization of expensive black box functions using radial basis functions, Journal of Global Optimization 31, 153–171] (CORS-RBF). We modify these algorithms so that they can generate multiple points for simultaneous evaluation in parallel. We compare the performance of the two parallel RBF methods with a parallel multistart derivative-based algorithm, a parallel multistart derivative-free trust-region algorithm, and a parallel evolutionary algorithm on eleven test problems and on a 6-dimensional groundwater bioremediation application. The results indicate that the two parallel RBF algorithms are generally better than the other three alternatives on most of the test problems. Moreover, the two parallel RBF algorithms have comparable performances on the test problems considered. Finally, we report good speedups for both parallel RBF algorithms when using a small number of processors.

Fast construction of constant bound functions for sparse polynomials

A new method for the representation and computation of Bernstein coefficients of multivariate polynomials is presented. It is known that the coefficients of the Bernstein expansion of a given polynomial over a specified box of interest tightly bound the range of the polynomial over the box. The traditional approach requires that all Bernstein coefficients are computed, and their number is often very large for polynomials with moderately-many variables. The new technique detailed represents the coefficients implicitly and uses lazy evaluation so as to render the approach practical for many types of non-trivial sparse polynomials typically encountered in global optimization problems; the computational complexity becomes nearly linear with respect to the number of terms in the polynomial, instead of exponential with respect to the number of variables. These range-enclosing coefficients can be employed in a branch-and-bound framework for solving constrained global optimization problems involving polynomial functions, either as constant bounds used for box selection, or to construct affine underestimating bound functions. If such functions are used to construct relaxations for a global optimization problem, then sub-problems over boxes can be reduced to linear programming problems, which are easier to solve. Some numerical examples are presented and the software used is briefly introduced.

Optimal Design of Piezoelectric Transformers: A Rational Approach Based on an Analytical Model and a Deterministic Global Optimization

This paper deals with a deterministic and rational way to design piezoelectric transformers in radial mode. The proposed approach is based on the study of the inverse problem of design and on its reformulation as a mixed constrained global optimization problem. The methodology relies on the association of the analytical models for describing the corresponding optimization problem and on an exact global optimization software, named IBBA and developed by the second author to solve it. Numerical experiments are presented and compared in order to validate the proposed approach.

A filled function method for constrained global optimization

In this paper, a filled function method for solving constrained global optimization problems is proposed. A filled function is proposed for escaping the current local minimizer of a constrained global optimization problem by combining the idea of filled function in unconstrained global optimization and the idea of penalty function in constrained optimization. Then a filled function method for obtaining a global minimizer or an approximate global minimizer of the constrained global optimization problem is presented. Some numerical results demonstrate the efficiency of this global optimization method for solving constrained global optimization problems.

Differential Evolution Strategy for Constrained Global Optimization and Application to Practical Engineering Problems

In this paper, some modifications to the differential evolution strategy are proposed for the constrained global optimization problem. The scaling factor F is varied randomly within some range and an auxiliary set is employed to enhance the diversity of the population. The newly generated trial vector is compared with the nearest parent. The simple feasibility rule is used to treat the constraints. The effectiveness of the proposed method is demonstrated by solving some practical engineering problems. Finally, the method was applied to the robust design of a gas circuit breaker to increase the small current interruption performance

A particle swarm pattern search method for bound constrained global optimization

In this paper we develop, analyze, and test a new algorithm for the global minimization of a function subject to simple bounds without the use of derivatives. The underlying algorithm is a pattern search method, more specifically a coordinate search method, which guarantees convergence to stationary points from arbitrary starting points. In the optional search phase of pattern search we apply a particle swarm scheme to globally explore the possible nonconvexity of the objective function. Our extensive numerical experiments showed that the resulting algorithm is highly competitive with other global optimization methods also based on function values.