Integral Global Optimization Research Articles

Extremely adaptive image retrieval scheme employing an optimized wavelet technique intended for characterization maps

The demand for adaptive image retrieval is still an active research area, particularly in a dynamic environment. Erstwhile retrieval schemes ensure adaptivity by tuning the same image basis with wavelet transforms, in accordance with user’s significance. To enhance adaptivity and improve the retrieval performance, an Extremely Adaptive Image Retrieval (EAIR) scheme is presented that associates each image with different wavelet basis. This objective is achieved by building Characterization maps from wavelet coefficients of images (query and target) using higher order standardized moments of the Gamma function. The resulting maps are approximated by Volterra Series and later, mathematically programmed by Integral Global Optimization (IGO) algorithm for wavelet adaptation. Finally, the best wavelet filter for each query image is fitted using the Multivariate Adaptive Regression Splines (MARS). Characterization Maps rendered by EAIR achieves 60% reduction in Relative Approximation Error (RAE) with 11% decrease in query time observed under diverse dataset. Also, relative Precision-Recall (P-R), Precision at 5 (P5) analyses reveals a significant retrieval improvement of 9.52%, 1.35%, 1.12%, 8.07% by EAIR on Caltech, Messidor, AT&T, Vistex respectively.

A modified integral global optimization method and its asymptotic convergence

In this paper, we propose a new integral global optimization algorithm for finding the solution of continuous minimization problem, and prove the asymptotic convergence of this algorithm. In our modified method we use variable measure integral, importance sampling and main idea of the cross-entropy method to ensure its convergence and efficiency. Numerical results show that the new method is very efficient in some challenging continuous global optimization problems.

Existence and optimality of accessible and approximatable minimizers of quasi upper robust functions

The concepts of robustness of sets and functions were proposed for the theory of integral global optimization which ensure that a robust minimizer can be approximated by a sequence of points at which the objective function is continuous. These concepts are generalized in this paper and global minimization of quasi upper robust functions is investigated. With the integral optimality conditions of global minimum, we examine existence of robust, accessible, and approximatable minimizers of discontinuous functions.

A conceptual method for solving generalized semi-infinite programming problems via global optimization by exact discontinuous penalization

We consider a generalized semi-infinite programming problem (GSIP) with one semi-infinite constraint where the index set depends on the variable to be minimized. Keeping in mind the integral global optimization method of Zheng and Chew and its modifications we would like to outline theoretical considerations for determining coarse approximations of a solution of (GSIP) via global optimization of an exact discontinuous penalty approach. We consider an auxiliary parametric semi-infinite programming problem and the behavior of its marginal functional. In so doing we extend the theory of robust analysis to study robustness of marginal functions and robustness of set valued mappings with given structures.

Modified integral-level set method for the constrained solving Global Optimization

The constrained global optimization problem being considered, a modified integral-level set method was illustrated based on Chew-Zheng’s paper on Integral Global Optimization and Wu’s paper on Implementable Algorithm Convergence of Modified Integral-Level Set Method for Global Optimization Problem. It has two characters: 1) Each phase must construct a new function which has the same global optimal value as that of primitive objective function; 2) Comparing it with Zheng’s method, solving level set procedure is avoided. An implementable algorithm also is given and it is proved that this algorithm is convergent.

Comparison of public-domain software for black box global optimization*

We instance our experience with six public-domain global optimization software products and report comparative computational results obtained on a set of eleven test problems. The techniques used by the software under study include integral global optimization, genetic algorithms, simulated annealing, clustering, random search, continuation, Bayesian, tunneling, and multi-level methods. The test set contains practical problems: least median of squares regression, protein folding, and multidimensional scaling. These include non-differentiable, and also discontinuous objective functions, some with an exponential number of local minima. The dimension of the search space ranges from 1 to 20. We evaluate the software in view of engineers addressing black box global optimization problems, i.e. problems with an objective function whose explicit form is unknown and whose evaluation is costly. Such an objective function is common in industry. It is for instance given under the form of computer programmes involving a simulation

Global optimum shape design

An optimum shape design problem can be formulated as a minimization problem of a functional subject to certain constraints. Usually, it is nonlinear and nonconvex. Conventional optimization techniques are gradient-based, they highly depend on the initial design, and are difficult to be applied to find a global solution. Integral global optimization algorithm is proposed to solve optimum shape design problems. Three design examples are given to illustrate the effectiveness of the algorithm.

Long cycles and neighborhood union in 1-tough graphs with large degree sums

The ideas of robust sets, robust functions and robustness of general set-valued maps were introduced by Chew and Zheng [7, 26], and further developed by Shi, Zheng, Zhuang [18, 19, 20], Phu, Hoffmann and Hichert [8, 9, 10, 17] to weaken up the semi-continuity requirements of certain global optimization algorithms. Robust analysis, along with measure theory, has well served as the basis for the integral global optimization method (IGOM) (Chew and Zheng [7]). Hence, we have attempted to extend the robust analysis of Zheng et al. to that of robustness of set-valued maps with given structures and marginal value functions. We are also of strong conviction that the results of our investigation could open a way to apply the IGOM for the numerical treatment of some class of parametric optimization problems, when global optima are required.

Integral global minimization: Algorithms, implementations and numerical tests

The theoretical foundation of integral global optimization has become widely known and well accepted [4],[24],[25]. However, more effort is needed to demonstrate the effectiveness of the integral global optimization algorithms. In this work we detail the implementation of the integral global minimization algorithms. We describe how the integral global optimization method handles nonconvex unconstrained or box constrained, constrained or discrete minimization problems. We illustrate the flexibility and the efficiency of integral global optimization method by presenting the performance of algorithms on a collection of well known test problems in global optimization literature. We provide the software which solves these test problems and other minimization problems. The performance of the computations demonstrates that the integral global algorithms are not only extremely flexible and reliable but also very efficient.

Discontinuous robust mappings are approximatable

The concepts of robustness of sets and and functions were introduced to form the foundation of the theory of integral global optimization. A set A A of a topological space X X is said to be robust iff cl A = cl {\text {cl}}A = {\text {cl}} int A A . A mapping f : X → Y f:X \to Y is said to be robust iff for each open set U Y {U_Y} of Y Y , f − 1 ( U Y ) {f^{ - 1}}({U_Y}) is robust. We prove that if X X is a Baire space and Y Y satisfies the second axiom of countability, then a mapping f : X → Y f:X \to Y is robust iff it is approximatable in the sense that the set of points of continuity of f f is dense in X X and that for any other point x ∈ X x \in X , ( x , f ( x ) ) (x,f(x)) is the limit of { ( x α , f ( x α ) ) } \{ ({x_\alpha },f({x_\alpha }))\} , where for all α \alpha , x α {x_\alpha } is a continuous point of f f . This result justifies the notion of robustness.

Integral global optimization method for solution of nonlinear complementarity problems

The mapping in a nonlinear complementarity problem may be discontinuous. The integral global optimization algorithm is proposed to solve a nonlinear complementarity problem with a robust piecewise continuous mapping. Numerical examples are given to illustrate the effectiveness of the algorithm.

Massively parallel structural design using stochastic optimization and mixed neuralnet/finite element analysis methods

The title study is performed on the massively parallel processing (MPP) environment of Connection Machine (CM) computers using truss structural sizing design problems as example design problems. In this design optimization procedure, only the displacement solution is replaced by that based on neural net technology (under a given set of cross-sectional size parameters (e.g., areas) in the MPP finite element structural reanalysis). This structural reanalysis procedure, together with a vastly improved and parallelized version of the integral global optimization (IGO) stochastic algorithm, IIGO, forms the present MPP structural design methodology. In addition, a procedure to correct the final optimal design for constraint violation or too-conservatively satisfied constraint condition caused by inaccuracy of the NN (neural network) analysis model is also formulated. Evaluation of the numerical performance of the developed computational algorithm set, capability, and strategy is made, primarily on the Connection Machine CM-2 model computer by performing three neural-network-based truss structural reanalysis/minimum weight design problems.

Set-Valued Robust Mappings and Approximatable Mappings

The concepts of robustness of sets and mappings were proposed for the theory of integral global optimization. This paper discusses the robustness of a set-valued mapping, associating with the stability problem. We extend the equivalence between the robustness and the approximatability to the case of a set-valued mapping. We deal also with "bi-robustness" and "in-robustness."

Integral global optimization method in statistical applications

Many statistical computations need a minimization method which can find global minima of a nonconvex, nonsmooth and even discontinuous objective function. Conventional optimization methods hardly serve that. The integral optimization method has been developed to satisfy such requirements. In this paper, we apply this method to several problems in statistics, as nonlinear regression, maximum likelihood estimation for three-parameter Weibull distribution, mixture densities and sampling, and show that the results obtained by a global minimization method are better than those by a gradient-based one.

Finite dimensional approximation to solutions of minimization problems in functional spaces

In this paper we consider minimization problems whose objectives are defined on functional spaces. The integral global optimization technique is applied to characterize a global minimum as the limit of a sequence of approximating solutions on finite dimensional subspaces. Necessary and sufficient optimality conditions are presented. A variable measure algorithm is proposed to find such approximating solutions. Examples are presented to illustrate the variable measure method

Integral global optimization method for nonlinear games

A derivative-free method for global solution of continuous nonlinear games is proposed. The method is based on the integral global optimization algorithm for mathematical programming and it does not employ gradient-based techniques nor the notion of convexity. The existence of a saddle point is not assumed a priori. Instead, the method delivers the global solutions for both players, and, if global saddle points exist, it yields the value of the game and the global saddle set. Several computational schemes are proposed and nonlinear games with uncertainties are considered. The ideas are illustrated by examples.

Integral global optimization method for differential games with application to pursuit-evasion games

A variation-free method for global solution of differential games is proposed. The method is based on the integral global optimization algorithm and it does not employ variation-based techniques nor the notion of convexity. The existence of a saddle-point is not assumed a priori. Instead, the method delivers the global minimax and maximin solutions for both players. A necessary and sufficient global minimax condition is presented. An application is made to the solution of nonlinear pursuit-evasion games. The ideas are illustrated by examples.