Parameter sensitivity study of the Nelder-Mead Simplex Method

This paper presents a parameter sensitivity study of the Nelder-Mead Simplex Method for unconstrained optimization. Nelder-Mead Simplex Method is very easy to implement in practice, because it does not require gradient computation; however, it is very sensitive to the choice of initial points selected. Fan-Zahara conducted a sensitivity study using a select set of test cases and suggested the best values for the parameters based on the highest percentage rate of successful minimization. Begambre-Laier used a strategy to control the Particle Swarm Optimization parameters based on the Nelder Mead Simplex Method in identifying structural damage. The main purpose of the paper is to extend their parameter sensitivity study to better understand the parameter’s behavior. The comprehensive parameter sensitivity study was conducted on seven test functions: B2, Beale, Booth, Wood, Rastrigin, Rosenbrock and Sphere Functions to search for common patterns and relationships each parameter has in producing the optimum solution. The results show important relations of the Nelder-Mead Simplex parameters: reflection, expansion, contraction, and Simplex size and how they impact the optimum solutions. This study is crucial, because better understanding of the parameters behavior can motivate current and future research using Nelder-Mead Simplex in creating an intelligent algorithm, which can be more effective, efficient, and save computational time.     

Discrete particle swarm optimization based on estimation of distribution for terminal assignment problems

Terminal assignment problem (TEAP) is to determine minimum cost links to form a network by connecting a given set of terminals to a given collection of concentrators. This paper presents a novel discrete particle swarm optimization (PSO) based on estimation of distribution (EDA), named DPSO-EDA, for TEAP. EDAs sample new solutions from a probability model which characterizes the distribution of promising solutions in the search space at each generation. The DPSO-EDA incorporates the global statistical information collected from personal best solutions of all particles into the PSO, and therefore each particle has comprehensive learning and search ability. In the DPSO-EDA, a modified constraint handling method based on Hopfield neural network (HNN) is also introduced to fit nicely into the framework of the PSO and thus utilize the merit of the PSO. The DPSO-EDA adopts the asynchronous updating scheme. Further, the DPSO-EDA is applied to a problem directly related to TEAP, the task assignment problem (TAAP), in order to show that the DPSO-EDA can be generalized to other related combinatorial optimization problems. Simulation results on several problem instances show that the DPSO-EDA is better than previous methods.     

A heuristic approach for packing identical rectangles in convex regions

In this paper we propose a heuristic approach for the problem of packing equal rectangles within a convex region. The approach is based on an Iterated Local Search scheme, in which the key step is the perturbation move. Different perturbation moves, both combinatorial and continuous ones, are proposed and compared through extensive computational experiments on a set of test instances. The overall results are quite encouraging.     

Some algebraic methods for solving multiobjective polynomial integer programs

Multiobjective discrete programming is a well-known family of optimization problems with a large spectrum of applications. The linear case has been tackled by many authors during the past few years. However, the polynomial case has not been studied in detail due to its theoretical and computational difficulties. This paper presents an algebraic approach for solving these problems. We propose a methodology based on transforming the polynomial optimization problem to the problem of solving one or more systems of polynomial equations and we use certain Gröbner bases to solve these systems. Different transformations give different methodologies that are theoretically stated and compared by some computational tests via the algorithms that they induce.     

Shape optimization for tumor location

In non-invasive thermal diagnostics, accurate correlations between the thermal image at skin surface and interior human physiology are desired. In this work, an estimation methodology to determine unknown geometrical parameters of an embedded tumor is proposed. We define a functional that represents the mismatch between a measured experimental temperature profile, which may be obtained by infrared thermography on the skin surface, and the solution of an appropriate boundary problem. This functional is related to the geometrical parameters through the solution of the boundary problem, in such a way that finding the minimum of this functional form also means finding the unknown geometrical parameters of the embedded tumor. Sensitivity analysis techniques coupled with the adjoint method were considered to compute the shape derivative of the functional. Then, a nonmonotone spectral projected gradient method was implemented to solve the optimization problem of finding the optimal geometric parameters.     

Tabu search-based metaheuristic algorithm for software system reliability problems

This paper presents a new metaheuristic-based algorithm for complex reliability problems. The algorithm effectively uses features of the Tabu Search paradigm, with special emphasis on the exploitation of memory-based mechanisms. It balances intensification with diversification via the use of short-term and long-term memory. The algorithm has been thoroughly tested on benchmark problems from the literature as well as on a pool of random generated instances of very large scale software systems. The proposed algorithm proves to be robust with respect to its parameters and it is especially suited for very large scale instances of the reliability problem, when exact approaches are doomed to fail.     

Learning a color distance metric for region-based image segmentation

In this paper we describe an experiment where we studied empirically the application of a learned distance metric to be used as discrimination function for an established color image segmentation algorithm. For this purpose we chose the Mumford–Shah energy functional and the Mahalanobis distance metric. The objective was to test our approach in an objective and quantifiable way on this specific algorithm employing this particular distance model, without making generalization claims. The empirical validation of the results was performed in two experiments: one applying the resulting segmentation method on a subset of the Berkeley Image Database, an exemplar image set possessing ground-truths and validating the results against the ground-truths using two well-known inter-cluster validation methods, namely, the Rand and BGM indexes, and another experiment using images of the same context divided into training and testing set, where the distance metric is learned from the training set and then applied to segment all the images. The obtained results suggest that the use of the specified learned distance metric provides better and more robust segmentations, even if no other modification of the segmentation algorithm is performed.     

Self-consistent gradient flow for shape optimization

We present a model for image segmentation and describe a gradient-descent method for level-set based shape optimization. It is commonly known that gradient-descent methods converge slowly due to zig–zag movement. This can also be observed for our problem, especially when sharp edges are present in the image. We interpret this in our specific context to gain a better understanding of the involved difficulties. One way to overcome slow convergence is the use of second-order methods. For our situation, they require derivatives of the potentially noisy image data and are thus undesirable. Hence, we propose a new method that can be interpreted as a self-consistent gradient flow and does not need any derivatives of the image data. It works very well in practice and leads to a far more efficient optimization algorithm. A related idea can also be used to describe the mean-curvature flow of a mean-convex surface. For this, we formulate a mean-curvature Eikonal equation, which allows a numerical propagation of the mean-curvature flow of a surface without explicit time stepping.     

Design optimization of resource combination for collaborative logistics network under uncertainty

Collaborative logistics networks (CLNs) are considered to be an effective organizational form for business cooperation that provides high stability and low cost. One common key issue regarding CLN resource combination is the network design optimization problem under discrete uncertainty (DU-CLNDOP). Operational environment changes and information uncertainty in network designs, due to partner selection, resource constrains and network robustness, must be effectively controlled from the system perspective. Therefore, a general two-stage quantitative framework that enables decision makers to select the optimal network design scheme for CLNs under uncertainty is proposed in this paper. Phase 1 calculates the simulation result of each hypothetical scenario of CLN resource combination using the expected value model with robust constraints. Phase 2 selects the optimal network design scheme for DU-CLNDOP using the orthogonal experiment design method. The validity of the model and method are verified via an illustrative example.