A New Version of the Price's Algorithm for Global Optimization

arXiv: Optimization and Control, 2020

In this paper, a sequential search method for finding the global minimum of an objective function is presented, The descent gradient search is repeated until the global minimum is obtained. The global minimum is located by a process of finding progressively better local minima. We determine the set of points of intersection between the curve of the function and the horizontal plane which contains the local minima previously found. Then, a point in this set with the greatest descent slope is chosen to be a initial point for a new descent gradient search. The method has the descent property and the convergence is monotonic. To demonstrate the effectiveness of the proposed sequential descent method, several non-convex multidimensional optimization problems are solved. Numerical examples show that the global minimum can be sought by the proposed method of sequential descent.

IEEE Transactions on Systems, Man, and Cybernetics, 2000

An efficient and practical solution to a class of global function optimization is proposed. The new algorithm consists of a stochastic search of initial guesses and a gradient-based solutionfinding algorithm. The key idea is to introduce an auxiliary cost function that can indicate whether the gradient-based solutionfinding process goes toward a global minimum of the cost function and that helps us to prevent the process from going to local minima. Simulation examples are used to show the mechanism, power, and restrictions of the approach.

Journal of Global Optimization, 2010

A large number of algorithms introduced in the literature to find the global minimum of a real function rely on iterative executions of searches of a local minimum. Multistart, tunneling and some versions of simulated annealing are methods that produce well-known procedures. A crucial point of these algorithms is to decide whether to perform or not a new local search. In this paper we look for the optimal probability value to be set at each iteration so that by moving from a local minimum to a new one, the average number of function evaluations evals is minimal. We find that this probability has to be 0 or 1 depending on the number of function evaluations required by the local search and by the size of the level set at the current point. An implementation based on the above result is introduced. The values required to calculate evals are estimated from the history of the algorithm at running time. The algorithm has been tested both for sample problems constructed by the GKLS package and for problems often used in the literature. The outcome is compared with recent results.

Journal of Computational Chemistry, 1997

A new algorithm is presented for the location of the global minimum of a multiple minima problem. It begins with a series of randomly placed probes in phase space, and then uses an iterative Gaussian redistribution of the worst probes into better regions of phase space until all probes converge to a single point. The method quickly converges, does not require derivatives, and is resistant to becoming trapped in local minima. Comparison of this algorithm with others using a standard test suite demonstrates that the number of function calls has been decreased conservatively by a factor of about three with the same degree of accuracy. A sample problem of a system of seven Lennard᎐Jones particles is presented as a concrete example.

Journal of Global Optimization, 2006

This paper presents a general approach that combines global search strategies with local search and attempts to find a global minimum of a real valued function of n variables. It assumes that derivative information is unreliable; consequently, it deals with derivative free algorithms, but derivative information can be easily incorporated. This paper presents a nonmonotone derivative free algorithm and shows numerically that it may converge to a better minimum starting from a local nonglobal minimum. This property is then incorporated into a random population to globalize the algorithm. Convergence to a zero order stationary point is established for nonsmooth convex functions, and convergence to a first order stationary point is established for strictly differentiable functions. Preliminary numerical results are encouraging. A Java implementation that can be run directly from the Web allows the interested reader to get a better insight of the performance of the algorithm on several standard functions. The general framework proposed here, allows the user to incorporate variants of well known global search strategies.

2004

An implementation problem of an algorithm for one-dimensional global optimization in the presence of noise is considered. The approach is based on modelling the objective function as a standard Wiener process. The testing results of the implemented algorithm are presented using several well known test functions. The efficiency of the algorithm is evaluated for different levels of noise.

2013

A procedure for generating non-differentiable, continuously differentiable, and twice continuously differentiable classes of test functions for multiextremal multidimensional box-constrained global optimization and a corresponding package of C subroutines are presented. Each test class consists of 100 functions. Test functions are generated by defining a convex quadratic function systematically distorted by polynomials in order to introduce local minima. To determine a class, the user defines the following parameters: (i) problem dimension, (ii) number of local minima, (iii) value of the global minimum, (iv) radius of the attraction region of the global minimizer, (v) distance from the global minimizer to the vertex of the quadratic function. Then, all other necessary parameters are generated randomly for all 100 functions of the class. Full information about each test function including locations and values of all local minima is supplied to the user. Partial derivatives are also generated where possible.

Journal of Heuristics, 2001

In this paper, uego, a new general technique for accelerating and/or parallelizing existing search methods is suggested. The skeleton of the algorithm is a parallel hill climber. The separate hill climbers work in restricted search regions (or clusters) of the search space. The volume of the clusters decreases as the search proceeds which results in a cooling effect similar to simulated annealing. Besides this, uego can be effectively parallelized; the communication between the clusters is minimal. The purpose of this communication is to ensure that one hill is explored only by one hill climber. uego makes periodic attempts to find new hills to climb. Empirical results are also presented which include an analysis of the effects of the user-given parameters and a comparison with a hill climber and a ga.

2006

In this paper we present a novel optimization algorithm that estimates gradients over regions to search for optima of a non-convex function on both a local and global scale. The proposed architecture is based on three concepts: using the memory of previously evaluated points, multiresolutional search, and the estimation of gradients at these different resolutions to direct the search. This multiresolution estimated gradient architecture (MEGA) shows promise to perform competitively when compared to standard global searches. Comparisons on the Rosenbrock, Griewank, and sinusoidal test functions show that MEGA can converge faster than particle swarm optimization, particularly as dimensionality of a problem increases.

Journal of Global Optimization, 2004

In this paper, a hybrid descent method, consisting of a simulated annealing algorithm and a gradient-based method, is proposed. The simulated annealing algorithm is used to locate descent points for previously converged local minima. The combined method has the descent property and the convergence is monotonic. To demonstrate the effectiveness of the proposed hybrid descent method, several multi-dimensional non-convex optimization problems are solved. Numerical examples show that global minimum can be sought via this hybrid descent method.