Nonsmooth Global Optimization Research Articles

A HYBRID DIFFERENTIAL EVOLUTION FOR NON-SMOOTH OPTIMIZATION PROBLEMS

Solving high dimentional, multimodal, non-smooth global optimization problems faces challenges concerning quality of solution, computational costs or even the impossibility of solving the problem. Evolutionary algorithms, in particular, differential evolution algorithm proved itself as good method of global optimization. On the other side, approach based on subgradient methods are good for optimizing non-smooth functions. Combination of these two approaches enables to improve the quality of the algorithm, using the best features of both methods. In this paper, a new hybrid evolutionary approach based on differential evolution and subgradient algorithm as the local search procedure is proposed. Behavior of the proposed SSGDE algorithm was studied in a numerical experiment on three groups of generated tests. Comparison of the new hybrid algorithm with the pure DE approach showed the advantage of the SSGDE. It has been experimentally established that the proposed method finds the global minimum in the best way for all considered dimensions of the problem with respect to the differential evolution method. The SSGDE algorithm showed the best results with a significant increase in the number of functions.

A New Filled Function for Non-smooth Global Optimization and Its Applications

This paper aims to discover a global optima for non-smooth optimal issues by designing a novel filled function, which, regarded as a bridge delivering one minimizer to another minimizer, contains only one parameter for easy adjustment. Theoretically, the natures of the considered filled function are conducted as well as a new algorithm is presented. Finally, certain experimental calculations are presented to illustrate the performance of the constructed algorithm.

New Auxiliary Function with Properties in Nonsmooth Global Optimization for Melanoma Skin Cancer Segmentation.

In this paper, an algorithm is introduced to solve the global optimization problem for melanoma skin cancer segmentation. The algorithm is based on the smoothing of an auxiliary function that is constructed using a known local minimizer and smoothed by utilising Bezier curves. This function achieves all filled function properties. The proposed optimization method is applied to find the threshold values in melanoma skin cancer images. The proposed algorithm is implemented on PH2, ISBI2016 challenge, and ISBI 2017 challenge datasets for melanoma segmentation. The results show that the proposed algorithm exhibits high accuracy, sensitivity, and specificity compared with other methods.

A modified tunneling function method for non-smooth global optimization and its application in artificial neural network

For solving a class of non-smooth unconstrained global optimization problems, we present a novel definition of the modified tunneling function which combines the characters of tunneling function and filled function, and then give a one-parameter modified tunneling function. Issues covered in the presented work include: theoretical properties, solution algorithms and numerical experiments. Furthermore, an improved artificial neural network hydrological forecasting method using the modified tunneling function is also reported. The preliminary experiment results confirm that the proposed approach is promising.

A CARTopt method for bound constrained global optimization

A stochastic method for bound constrained global optimization is described. The method can be applied to objective functions which are nonsmooth or even discontinuous. The algorithm forms a partition on the search region using classification and regression trees (CART), which defines desirable subsets where the objective function is relatively low. Further samples are drawn directly from these low subsets before a new partition is formed. Alternating between sampling and partition phases provides an effective method for nonsmooth global optimization. The sequence of iterates generated by the algorithm is shown to converge to a global minimizer of the objective function with probability one under mild conditions. Non-probabilistic results are also give when random sampling is replaced with samples taken from the Halton sequence. Numerical results are presented for both smooth and nonsmooth problems and show that the method is effective and competitive in practice. doi:10.1017/S1446181113000412

A CARTOPT METHOD FOR BOUND-CONSTRAINED GLOBAL OPTIMIZATION

AbstractA stochastic algorithm for bound-constrained global optimization is described. The method can be applied to objective functions that are nonsmooth or even discontinuous. The algorithm forms a partition on the search region using classification and regression trees (CART), which defines a region where the objective function is relatively low. Further points are drawn directly from the low region before a new partition is formed. Alternating between partition and sampling phases provides an effective method for nonsmooth global optimization. The sequence of iterates generated by the algorithm is shown to converge to an essential global minimizer with probability one under mild conditions. Nonprobabilistic results are also given when random sampling is replaced with points taken from the Halton sequence. Numerical results are presented for both smooth and nonsmooth problems and show that the method is effective and competitive in practice.

Derivative-free optimization and neural networks for robust regression

Large outliers break down linear and nonlinear regression models. Robust regression methods allow one to filter out the outliers when building a model. By replacing the traditional least squares criterion with the least trimmed squares (LTS) criterion, in which half of data is treated as potential outliers, one can fit accurate regression models to strongly contaminated data. High-breakdown methods have become very well established in linear regression, but have started being applied for non-linear regression only recently. In this work, we examine the problem of fitting artificial neural networks (ANNs) to contaminated data using LTS criterion. We introduce a penalized LTS criterion which prevents unnecessary removal of valid data. Training of ANNs leads to a challenging non-smooth global optimization problem. We compare the efficiency of several derivative-free optimization methods in solving it, and show that our approach identifies the outliers correctly when ANNs are used for nonlinear regression.

Global minimization of non-smooth unconstrained problems with filled function

For smooth or non-smooth unconstrained global optimization problems, an one parameter filled function is derived to identify their global optimizers or approximately global optimizers. The theoretical properties of the proposed function are investigated. Based on the filled function, an algorithm is designed for solving unconstrained global optimization problems. The algorithm consists of two phases: local minimization and filling. The former is intended to minimize the objective function and obtain a local optimizer, the latter aims to find a better initial point for the first phase. Numerical experimentation is also provided. The preliminary computational results confirm that the proposed filled function approach is promising.

A Novel Filled Function for Solving Non-smooth Global Optimization Problem

A Novel Filled Function for Solving Non-smooth Global Optimization Problem

A Generalized Filled Function for Non-Smooth Constrained Global Optimization

A generalized filled function method is developed in this paper to solve nonsmooth constrained global optimization problem. Theoretical properties of the proposed function is investigated, and a solution algorithm is proposed. Numerical experiments are also reported in this paper. The preliminary computational results show that the proposed method is promising.

A one-parameter filled function for nonsmooth global optimization and its application

This paper presents a modified definition of the filled function for finding a global minimizer of a nonsmooth function on a closed bounded set, and then give a one-parameter filled function. Theoretical and numerical properties of the proposed filled function are investigated and a corresponding solution algorithm is proposed. The proposed filled function’s parameter is easier to be appropriately chosen than previous functions in literatures. Numerical results obtained indicate the efficiency of the proposed filled function method. An improved fingerprint recognition method using global filled function is also reported.

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 multidimensional descent method for global optimization

This article presents a new multidimensional descent method for solving global optimization problems with box-constraints. This is a hybrid method where local search method is used for a local descent and global search is used for further multidimensional search on the subsets of intersection of cones generated by the local search method and the feasible region. The discrete gradient method is used for local search and the cutting angle method is used for global search. Two- and three-dimensional cones are used for the global search. Such an approach allows one, as a rule, to escape local minimizers which are not global ones. The proposed method is local optimization method with strong global search properties. We present results of numerical experiments using both smooth and non-smooth global optimization test problems. These results demonstrate that the proposed algorithm allows one to find a global or a near global minimizer.

A second-order pruning step for verified global optimization

We consider pruning steps used in a branch-and-bound algorithm for verified global optimization. A first-order pruning step was given by Ratz using automatic computation of a first-order slope tuple (Ratz, Automatic Slope Computation and its Application in Nonsmooth Global Optimization. Shaker Verlag, Aachen, 1998; J. Global Optim. 14: 365–393, 1999). In this paper, we introduce a second-order pruning step which is based on automatic computation of a second-order slope tuple. We add this second-order pruning step to the algorithm of Ratz. Furthermore, we compare the new algorithm with the algorithm of Ratz by considering some test problems for verified global optimization on a floating-point computer.

A new pruning test for finding all global minimizers of nonsmooth functions

A new pruning test is presented via interval slopes for finding all global minimizers of nonsmooth functions of several variables, based on an expansion scheme which can be used to achieve better interval slopes and enclosure of function ranges. The presented pruning test extends Ratz’s [A nonsmooth global optimization technique using slopes: one-dimensional case, J. Global Optimiz. 4 (1999) 365–393] pruning technique, which can be extensively applied in several variable nonsmooth global optimization as a accelerating device. This new pruning test is similar to the monotonicity test frequently used in interval methods for smooth problems, but it improves the monotonicity test. Numerical results show that the proposed global optimization algorithm with the new pruning test is better than that with the monotonicity test for smooth functions and superior to other interval algorithm for nonsmooth functions.

Applications of interval arithmetic in non-smooth global optimization

In this paper, an expansion scheme for general functions is presented, which can be used to achieve better enclosure of function ranges. In the context of interval branch-and-bound methods for non-smooth global optimization, a pruning technique is given via the expansion. An interval pruning test is established. This pruning test offers the possibility to cut away a large part of the currently investigated box by the optimization algorithm, which can be utilized as an accelerating device similar to the monotonicity test frequently used in interval methods for smooth problems. Numerical computation shows that the proposed method is reliable and can find all solutions of global optimization problem.

Canonical Dual Transformation Method and Generalized Triality Theory in Nonsmooth Global Optimization

This paper presents, within a unified framework, a potentially powerful canonical dual transformation method and associated generalized duality theory in nonsmooth global optimization. It is shown that by the use of this method, many nonsmooth/nonconvex constrained primal problems in \realn can be reformulated into certain smooth/convex unconstrained dual problems in \realm with m≤slant n and without duality gap, and some NP-hard concave minimization problems can be transformed into unconstrained convex minimization dual problems. The extended Lagrange duality principles proposed recently in finite deformation theory are generalized suitable for solving a large class of nonconvex and nonsmooth problems. The very interesting generalized triality theory can be used to establish nice theoretical results and to develop efficient alternative algorithms for robust computations.

A Nonsmooth Global Optimization Technique Using Slopes: The One-Dimensional Case

In this paper we introduce a pruning technique based on slopes in the context of interval branch-and-bound methods for nonsmooth global optimization. We develop the theory for a slope pruning step which can be utilized as an accelerating device similar to the monotonicity test frequently used in interval methods for smooth problems. This pruning step offers the possibility to cut away a large part of the box currently investigated by the optimization algorithm. We underline the new technique‘s efficiency by comparing two variants of a global optimization model algorithm: one equipped with the monotonicity test and one equipped with the pruning step. For this reason, we compared the required CPU time, the number of function and derivative or slope evaluations, and the necessary storage space when solving several smooth global optimization problems with the two variants. The paper concludes on the test results for several nonsmooth examples.