The simplex method as a global optimizer: A c-programming perspective

In this paper we give a brief account of the important role that the conventional simplex method of linear programming can play in global optimization, focusing on its collaboration with composite concave programming techniques. In particular, we demonstrate how rich and powerful the c-programming format is in cases where its parametric problem is a standard linear programming problem.     

A critical review of discrete filled function methods in solving nonlinear discrete optimization problems

Many real life problems can be modeled as nonlinear discrete optimization problems. Such problems often have multiple local minima and thus require global optimization methods. Due to high complexity of these problems, heuristic based global optimization techniques are usually required when solving large scale discrete optimization or mixed discrete optimization problems. One of the more recent global optimization tools is known as the discrete filled function method. Nine variations of the discrete filled function method in literature are identified and a review on theoretical properties of each method is given. Some of the most promising filled functions are tested on various benchmark problems. Numerical results are given for comparison.     

Optimization and data mining in medicine

Mathematical theory of optimization has found many applications in the area of medicine over the last few decades. Several data analysis and decision making problems in medicine can be formulated using optimization and data mining techniques. The significance of the mathematical models is greatly realized in the recent years owing to the growing technological capabilities and the large amounts of data available. In this paper, we attempt to give a brief overview of some of the most interesting applications of mathematical programming and data mining in medicine. In the overview, we include applications like radiation therapy treatment, microarray data analysis, and computational neuroscience.     

A metamodel-assisted evolutionary algorithm for expensive optimization

Expensive optimization aims to find the global minimum of a given function within a very limited number of function evaluations. It has drawn much attention in recent years. The present expensive optimization algorithms focus their attention on metamodeling techniques, and call existing global optimization algorithms as subroutines. So it is difficult for them to keep a good balance between model approximation and global search due to their two-part property. To overcome this difficulty, we try to embed a metamodel mechanism into an efficient evolutionary algorithm, low dimensional simplex evolution (LDSE), in this paper. The proposed algorithm is referred to as the low dimensional simplex evolution extension (LDSEE). It is inherently parallel and self-contained. This renders it very easy to use. Numerical results show that our proposed algorithm is a competitive alternative for expensive optimization problems.     

A metamodel-assisted evolutionary algorithm for expensive optimization

Expensive optimization aims to find the global minimum of a given function within a very limited number of function evaluations. It has drawn much attention in recent years. The present expensive optimization algorithms focus their attention on metamodeling techniques, and call existing global optimization algorithms as subroutines. So it is difficult for them to keep a good balance between model approximation and global search due to their two-part property. To overcome this difficulty, we try to embed a metamodel mechanism into an efficient evolutionary algorithm, low dimensional simplex evolution (LDSE), in this paper. The proposed algorithm is referred to as the low dimensional simplex evolution extension (LDSEE). It is inherently parallel and self-contained. This renders it very easy to use. Numerical results show that our proposed algorithm is a competitive alternative for expensive optimization problems.     

A global optimization approach for the linear two-level program

Linear two-level programming deals with optimization problems in which the constraint region is implicity determined by another optimization problem. Mathematical programs of this type arise in connection with policy problems to which the Stackelberg leader-follower game is applicable. In this paper, the linear two-level programming problem is restated as a global optimization problem and a new solution method based on this approach is developed. The most important feature of this new method is that it attempts to take full advantage of the structure in the constraints using some recent global optimization techniques. A small example is solved in order to illustrate the approach.The paper was completed while this author was visiting the Department of Mathematics of Linköping University.     

Random and Quasi-Random Linkage Methods in Global Optimization

In this paper a brief survey of recent developments in the field of stochastic global optimization methods will be presented. Most methods discussed fall in the category of two-phase algorithms, consisting in a global or exploration phase, obtained through sampling in the feasible domain, and a second or local phase, consisting of refinement of local knowledge, obtained through classical descent routines. A new class of methods is also introduced, characterized by the fact that sampling is performed through deterministic, well distributed, sample points. It is argued that for moderately sized problems this approach might prove more efficient than those based upon uniform random samples.     

Advances in Interval Methods for Deterministic Global Optimization in Chemical Engineering

In recent years, it has been shown that strategies based on an interval-Newton approach can be used to reliably solve a variety of nonlinear equation solving and optimization problems in chemical process engineering, including problems in parameter estimation and in the computation of phase behavior. These strategies provide a mathematical and computational guarantee either that all solutions have been located in an equation solving problem or that the global optimum has been found in an optimization problem. The primary drawback to this approach is the potentially high computational cost. In this paper, we consider strategies for bounding the solution set of the linear interval equation system that must be solved in the context of the interval-Newton method. Recent preconditioning techniques for this purpose are reviewed, and a new bounding approach based on the use of linear programming (LP) techniques is presented. Using this approach it is possible to determine the desired bounds exactly (within round out), leading to significant overall improvements in computational efficiency. These techniques will be demonstrated using several global optimization problems, with focus on problems arising in chemical engineering, including parameter estimation and molecular modeling. These problems range in size from under ten variables to over two hundred, and are solved deterministically using the interval methodology.     

Solving Planning and Design Problems in the Process Industry Using Mixed Integer and Global Optimization

This contribution gives an overview on the state-of-the-art and recent advances in mixed integer optimization to solve planning and design problems in the process industry. In some case studies specific aspects are stressed and the typical difficulties of real world problems are addressed. Mixed integer linear optimization is widely used to solve supply chain planning problems. Some of the complicating features such as origin tracing and shelf life constraints are discussed in more detail. If properly done the planning models can also be used to do product and customer portfolio analysis. We also stress the importance of multi-criteria optimization and correct modeling for optimization under uncertainty. Stochastic programming for continuous LP problems is now part of most optimization packages, and there is encouraging progress in the field of stochastic MILP and robust MILP. Process and network design problems often lead to nonconvex mixed integer nonlinear programming models. If the time to compute the solution is not bounded, there are already a commercial solvers available which can compute the global optima of such problems within hours. If time is more restricted, then tailored solution techniques are required.     

Primal and dual multi-objective linear programming algorithms for linear multiplicative programmes

Multiplicative programming problems (MPPs) are global optimization problems known to be NP-hard. In this paper, we employ algorithms developed to compute the entire set of nondominated points of multi-objective linear programmes (MOLPs) to solve linear MPPs. First, we improve our own objective space cut and bound algorithm for convex MPPs in the special case of linear MPPs by only solving one linear programme in each iteration, instead of two as the previous version indicates. We call this algorithm, which is based on Benson’s outer approximation algorithm for MOLPs, the primal objective space algorithm. Then, based on the dual variant of Benson’s algorithm, we propose a dual objective space algorithm for solving linear MPPs. The dual algorithm also requires solving only one linear programme in each iteration. We prove the correctness of the dual algorithm and use computational experiments comparing our algorithms to a recent global optimization algorithm for linear MPPs from the literature as well as two general global optimization solvers to demonstrate the superiority of the new algorithms in terms of computation time. Thus, we demonstrate that the use of multi-objective optimization techniques can be beneficial to solve difficult single objective global optimization problems.     

Stochastic Global Optimization: Problem Classes and Solution Techniques

There is a lack of a representative set of test problems for comparing global optimization methods. To remedy this a classification of essentially unconstrained global optimization problems into unimodal, easy, moderately difficult, and difficult problems is proposed. The problem features giving this classification are the chance to miss the region of attraction of the global minimum, embeddedness of the global minimum, and the number of minimizers. The classification of some often used test problems are given and it is recognized that most of them are easy and some even unimodal. Global optimization solution techniques treated are global, local, and adaptive search and their use for tackling different classes of problems is discussed. The problem of fair comparison of methods is then adressed. Further possible components of a general global optimization tool based on the problem classes and solution techniques is presented.     

Global optimality condition and fixed point continuation algorithm for non-Lipschitz ?_p regularized matrix minimization

Regularized minimization problems with nonconvex, nonsmooth, even non-Lipschitz penalty functions have attracted much attention in recent years, owing to their wide applications in statistics, control,system identification and machine learning. In this paper, the non-Lipschitz ?_p(0 p 1) regularized matrix minimization problem is studied. A global necessary optimality condition for this non-Lipschitz optimization problem is firstly obtained, specifically, the global optimal solutions for the problem are fixed points of the so-called p-thresholding operator which is matrix-valued and set-valued. Then a fixed point iterative scheme for the non-Lipschitz model is proposed, and the convergence analysis is also addressed in detail. Moreover,some acceleration techniques are adopted to improve the performance of this algorithm. The effectiveness of the proposed p-thresholding fixed point continuation(p-FPC) algorithm is demonstrated by numerical experiments on randomly generated and real matrix completion problems.     

New interval methods for constrained global optimization

Interval analysis is a powerful tool which allows to design branch-and-bound algorithms able to solve many global optimization problems. In this paper we present new adaptive multisection rules which enable the algorithm to choose the proper multisection type depending on simple heuristic decision rules. Moreover, for the selection of the next box to be subdivided, we investigate new criteria. Both the adaptive multisection and the subinterval selection rules seem to be specially suitable for being used in inequality constrained global optimization problems. The usefulness of these new techniques is shown by computational studies.     

Geomechanical parameters identification by particle swarm optimization and support vector machine

Back analysis is commonly used in identifying geomechanical parameters based on the monitored displacements. Conventional back analysis method is not capable of recognizing non-linear relationship involving displacements and mechanical parameters effectively. The new intelligent displacement back analysis method proposed in this paper is the combination of support vector machine, particle swarm optimization, and numerical analysis techniques. The non-linear relationship is efficiently represented by support vector machine. Numerical analysis is used to create training and testing samples for recognition of SVMs. Then, a global optimum search on the obtained SVMs by particle swarm optimization can lead to the geomechanical parameters identification effectively.     

Simulated annealing and object point processes: Tools for analysis of spatial patterns

This paper introduces a three-dimensional object point process—the Bisous model—that can be used as a prior for three-dimensional spatial pattern analysis. Maximization of likelihood or penalized-likelihood functions based on this model requires global optimization techniques, such as the simulated annealing algorithm. Theoretical properties of the model are discussed and the convergence of the proposed optimization method is proved. Finally, a simulation study is presented.     

Randomized algorithms in interval global optimization

This paper is a critical survey of the interval optimization methods aimed at computing global optima for multivariable functions. To overcome some drawbacks of traditional deterministic interval techniques, we outline some ways of constructing stochastic (randomized) algorithms in interval global optimization, in particular, those based on the ideas of random search and simulated annealing.     

Spectral bundle methods for non-convex maximum eigenvalue functions: first-order methods

Many challenging problems in automatic control may be cast as optimization programs subject to matrix inequality constraints. Here we investigate an approach which converts such problems into non-convex eigenvalue optimization programs and makes them amenable to non-smooth analysis techniques like bundle or cutting plane methods. We prove global convergence of a first-order bundle method for programs with non-convex maximum eigenvalue functions. Dedicated to R. T. Rockafellar on the occasion of his 70th anniversary     

离散优化与连续优化的复杂性概念

问题的复杂性概念起源于离散的图灵计算机理论的研究,在离散优化问题的研究中被广泛的接受.近期连续优化领域的很多文章中提及NP难这个概念.从而来对比介绍离散优化和连续优化研究中这两个概念的差异.     

Some transformation techniques with applications in global optimization

In this paper some transformation techniques, based on power transformations, are discussed. The techniques can be applied to solve optimization problems including signomial functions to global optimality. Signomial terms can always be convexified and underestimated using power transformations on the individual variables in the terms. However, often not all variables need to be transformed. A method for minimizing the number of original variables involved in the transformations is, therefore, presented. In order to illustrate how the given method can be integrated into the transformation framework, some mixed integer optimization problems including signomial functions are finally solved to global optimality using the given techniques.