Algorithm For Constrained Global Optimization Research Articles

A kriging-assisted bi-objective constrained global optimization algorithm for expensive constrained optimization problems

Computationally expensive constrained optimization problems are challenging owing to their high complexity and computational cost. To solve these problems efficiently, a kriging-assisted bi-objective constrained global optimization (BOCGO) algorithm is developed, where three phases with three bi-objective subproblems are performed. In phase I, the constraints are searched locally and globally to find the feasible region. Once a feasible region has been located, the two terms of the constrained expected improvement function are utilized to exploit and explore the feasible region in phase II. As the kriging models are accurate enough in the concerned region, a local search is processed to improve the optimal solution in phase III. The capability of the BOCGO algorithm is demonstrated by comparison with two classical and two state-of-the-art algorithms on 20 problems and an engineering simulation problem. The results show that the BOCGO algorithm performs better in more than three-fifths of problems, illustrating its effectiveness and robustness.

An efficient constrained global optimization algorithm with a clustering-assisted multiobjective infill criterion using Gaussian process regression for expensive problems

• An efficient constrained global optimization algorithm via the Gaussian process regression model is developed. • A clustering assisted multi-objective infill criterion is developed. • The accuracy of constraint model is considered. • The trade-off between objective and constraints is balanced. • The selection of sample points is adaptively. • The efficiency and robustness of the proposed approach are tested through 20 cases. Constrained optimization problems trouble engineers and researchers because of their high complexity and computational cost. When the objective function and constraints are both expensive black-box problems, there are many difficulties in solving them due to the unknown mathematical expressions and limited computational resources. To address these difficulties, we propose an efficient constrained global optimization algorithm. In the proposed algorithm, Gaussian process regression models are used to approximate the expensive objective function and constraints. Differential evolution (DE) is adopted to find the minimum value of the constrained lower confidence bounding (LCB). To further improve the accuracy of the Gaussian process regression models for the objective and constraints simultaneously, a clustering-assisted multiobjective infill criterion is proposed. The multiobjective infill criterion is utilized to balance the exploration between the objective and constraints. The clustering selection method is used to maintain the diversity of the sample points. The experimental results show that the proposed algorithm is better than or at least comparable to classic algorithms and other state-of-the-art algorithms

SCGOSR: Surrogate-based constrained global optimization using space reduction

Global optimization problems with computationally expensive objective and constraints are challenging. In this work, we present a new kriging-based constrained global optimization algorithm SCGOSR that can find global optima with fewer objective and constraint function evaluations. In SCGOSR, we propose a multi-start constrained optimization algorithm that can capture approximately local optimal points from kriging and select the promising ones for updating. In addition, according to two different penalty functions, two subspaces are created to construct local surrogate models and speed up the local search. Subspace1 is the neighborhood of the presented best solution, and Subspace2 is a region that covers several promising samples. The proposed multi-start constrained optimization is carried out alternately in Subspace1, Subspace2 and the global space. With iterations going on, kriging models of the costly objective and constraints are dynamically updated. In order to guarantee the balance between local and global search, the estimated mean square error of kriging is used to explore the unknown design space. Once SCGOSR gets stuck in a local valley, the algorithm will focus on the sparsely sampled regions. After comparison with 6 surrogate-based optimization algorithms on 13 representative cases, SCGOSR shows noticeable advantages in handling computationally expensive black-box problems.

Theoretical and Practical Convergence of a Self-Adaptive Penalty Algorithm for Constrained Global Optimization

This paper proposes a self-adaptive penalty function and presents a penalty-based algorithm for solving nonsmooth and nonconvex constrained optimization problems. We prove that the general constrained optimization problem is equivalent to a bound constrained problem in the sense that they have the same global solutions. The global minimizer of the penalty function subject to a set of bound constraints may be obtained by a population-based meta-heuristic. Further, a hybrid self-adaptive penalty firefly algorithm, with a local intensification search, is designed, and its convergence analysis is established. The numerical experiments and a comparison with other penalty-based approaches show the effectiveness of the new self-adaptive penalty algorithm in solving constrained global optimization problems.

A Kriging-based constrained global optimization algorithm for expensive black-box functions with infeasible initial points

In many engineering optimization problems, the objective and the constraints which come from complex analytical models are often black-box functions with extensive computational effort. In this case, it is necessary for optimization process to use sampling data to fit surrogate models so as to reduce the number of objective and constraint evaluations as soon as possible. In addition, it is sometimes difficult for the constrained optimization problems based on surrogate models to find a feasible point, which is the premise of further searching for a global optimal feasible solution. For this purpose, a new Kriging-based Constrained Global Optimization (KCGO) algorithm is proposed. Unlike previous Kriging-based methods, this algorithm can dispose black-box constrained optimization problem even if all initial sampling points are infeasible. There are two pivotal phases in KCGO algorithm. The main task of the first phase is to find a feasible point when there is no feasible data in the initial sample. And the aim of the second phase is to obtain a better feasible point under the circumstances of fewer expensive function evaluations. Several numerical problems and three design problems are tested to illustrate the feasibility, stability and effectiveness of the proposed method.

A Derivative-Free Algorithm for Constrained Global Optimization Based on Exact Penalty Functions

Constrained global optimization problems can be tackled by using exact penalty approaches. In a preceding paper, we proposed an exact penalty algorithm for constrained problems which combines an unconstrained global minimization technique for minimizing a non-differentiable exact penalty function for given values of the penalty parameter, and an automatic updating of the penalty parameter that occurs only a finite number of times. However, in the updating of the penalty parameter, the method requires the evaluation of the derivatives of the problem functions. In this work, we show that an efficient updating can be implemented also without using the problem derivatives, in this way making the approach suitable for globally solving constrained problems where the derivatives are not available. In the algorithm, any efficient derivative-free unconstrained global minimization technique can be used. In particular, we adopt an improved version of the DIRECT algorithm. In addition, to improve the performances, the approach is enriched by resorting to derivative-free local searches, in a multistart framework. In this context, we prove that, under suitable assumptions, for every global minimum point there exists a neighborhood of attraction for the local search. An extensive numerical experience is reported.

A Constrained Global Optimization Algorithm with Application to PI Robust Synthesis

We propose in this note a Global Optimization algorithm for multivariable constrained problems and apply it to the robust synthesis of PI controllers. The results we obtain on some test problems taken from the recent literature are reported.

A Stochastic Algorithm for Constrained Global Optimization

We present a stochastic algorithm to solve numerically the problem of finding the global minimizers of a real valued function subject to lower and upper bounds. This algorithm looks for the global minimizers following the paths of a suitable system of stochastic differential equations. Numerical experience on several test problems known in literature is shown.