Searching the solution space in constructive geometric constraint solving with genetic algorithms

Geometric problems defined by constraints have an exponential number of solution instances in the number of geometric elements involved. Generally, the user is only interested in one instance such that besides fulfilling the geometric constraints, exhibits some additional properties. Selecting a solution instance amounts to selecting a given root every time the geometric constraint solver needs to compute the zeros of a multi valuated function. The problem of selecting a given root is known as the Root Identification Problem. In this paper we present a new technique to solve the root identification problem. The technique is based on an automatic search in the space of solutions performed by a genetic algorithm. The user specifies the solution of interest by defining a set of additional constraints on the geometric elements which drive the search of the genetic algorithm. The method is extended with a sequential niche technique to compute multiple solutions. A number of case studies illustrate the performance of the method.Postprint (published version

Optimization with Discrete Simultaneous Perturbation Stochastic Approximation Using Noisy Loss Function Measurements

Discrete stochastic optimization considers the problem of minimizing (or maximizing) loss functions defined on discrete sets, where only noisy measurements of the loss functions are available. The discrete stochastic optimization problem is widely applicable in practice, and many algorithms have been considered to solve this kind of optimization problem. Motivated by the efficient algorithm of simultaneous perturbation stochastic approximation (SPSA) for continuous stochastic optimization problems, we introduce the middle point discrete simultaneous perturbation stochastic approximation (DSPSA) algorithm for the stochastic optimization of a loss function defined on a p-dimensional grid of points in Euclidean space. We show that the sequence generated by DSPSA converges to the optimal point under some conditions. Consistent with other stochastic approximation methods, DSPSA formally accommodates noisy measurements of the loss function. We also show the rate of convergence analysis of DSPSA by solving an upper bound of the mean squared error of the generated sequence. In order to compare the performance of DSPSA with the other algorithms such as the stochastic ruler algorithm (SR) and the stochastic comparison algorithm (SC), we set up a bridge between DSPSA and the other two algorithms by comparing the probability in a big-O sense of not achieving the optimal solution. We show the theoretical and numerical comparison results of DSPSA, SR, and SC. In addition, we consider an application of DSPSA towards developing optimal public health strategies for containing the spread of influenza given limited societal resources

VI Workshop on Computational Data Analysis and Numerical Methods: Book of Abstracts

The VI Workshop on Computational Data Analysis and Numerical Methods (WCDANM) is going to be held on June 27-29, 2019, in the Department of Mathematics of the University of Beira Interior (UBI), Covilhã, Portugal and it is a unique opportunity to disseminate scientific research related to the areas of Mathematics in general, with particular relevance to the areas of Computational Data Analysis and Numerical Methods in theoretical and/or practical field, using new techniques, giving especial emphasis to applications in Medicine, Biology, Biotechnology, Engineering, Industry, Environmental Sciences, Finance, Insurance, Management and Administration. The meeting will provide a forum for discussion and debate of ideas with interest to the scientific community in general. With this meeting new scientific collaborations among colleagues, namely new collaborations in Masters and PhD projects are expected. The event is open to the entire scientific community (with or without communication/poster)

Advances in System Identification and Stochastic Optimization

This work studies the framework of systems with subsystems, which has numerous practical applications, including system reliability estimation, sensor networks, and object detection. Consider a stochastic system composed of multiple subsystems, where the outputs are distributed according to many of the most common distributions, such as Gaussian, exponential and multinomial. In Chapter 1, we aim to identify the parameters of the system based on the structural knowledge of the system and the integration of data independently collected from multiple sources. Using the principles of maximum likelihood estimation, we provide the formal conditions for the convergence of the estimates to the true full system and subsystem parameters. The asymptotic normalities for the estimates and their connections to Fisher information matrices are also established, which are useful in providing the asymptotic or finite-sample confidence bounds. The maximum likelihood approach is then connected to general stochastic optimization via the recursive least squares estimation in Chapter 2. For stochastic optimization, we consider minimizing a loss function with only noisy function measurements and propose two general-purpose algorithms. In Chapter 3, the mixed simultaneous perturbation stochastic approximation (MSPSA) is introduced, which is designed for mixed variable (mixture of continuous and discrete variables) problems. The proposed MSPSA bridges the gap of dealing with mixed variables in the SPSA family, and unifies the framework of simultaneous perturbation as both the standard SPSA and discrete SPSA can now be deemed as two special cases of MSPSA. The almost sure convergence and rate of convergence of the MSPSA iterates are also derived. The convergence results reveal that the finite-sample bound of MSPSA is identical to discrete SPSA when the problem contains only discrete variables, and the asymptotic bound of MSPSA has the same order of magnitude as SPSA when the problem contains only continuous variables. In Chapter 4, the complex-step SPSA (CS-SPSA) is introduced, which utilizes the complex-valued perturbations to improve the efficiency of the standard SPSA. We prove that the CS-SPSA iterates converge almost surely to the optimum and achieve an accelerated convergence rate, which is faster than the standard convergence rate in derivative-free stochastic optimization algorithms

Simulation Optimization for Manufacturing System Design

A manufacturing system characterized by its stochastic nature, is defined by both qualitative and quantitative variables. Often there exists a situation when a performance measure such as throughput, work-in-process or cycle time of the system needs to be optimized with respect to some decision variables. It is generally convenient to express a manufacturing system in the form of an analytical model, to get the solutions as quickly as possible. However, as the complexity of the system increases, it gets more and more difficult to accommodate that complexity into the analytical model due to the uncertainty involved. In such situations, we resort to simulation modeling as an effective alternative.Equipment selection forms a separate class of problems in the domain of manufacturing systems. It assumes a high significance for capital-intensive industry, especially the semiconductor industry whose equipment cost comprises a significant amount of the total budget spent. For semiconductor wafer fabs that incorporate complex product flows of multiple product families, a reduction in the cycle time through the choice of appropriate equipment could result in significant profits. This thesis focuses on the equipment selection problem, which selects tools for the workstations with a choice of different tool types at each workstation. The objective is to minimize the average cycle time of a wafer lot in a semiconductor fab, subject to throughput and budget constraints. To solve the problem, we implement five simulation-based algorithms and an analytical algorithm. The simulation-based algorithms include the hill climbing algorithm, two gradient-based algorithms biggest leap and safer leap, and two versions of the nested partitions algorithm. We compare the performance of the simulation-based algorithms against that of the analytical algorithm and discuss the advantages of prior knowledge of the problem structure for the selection of a suitable algorithm

Solving the vehicle routing problem with stochastic demands using the cross-entropy method.

Abstract An alternate formulation of the classical vehicle routing problem with stochastic demands (VRPSD) is considered. We propose a new heuristic method to solve the problem. The algorithm is a modified version of the so-called Cross-Entropy method, which has been proposed in the literature as a heuristic for deterministic combinatorial optimization problems based upon concepts of rare-event simulation. In our version of the method, the objective function is computed using Monte-Carlo simulations at each point in the domain and the modified CrossEntropy heuristic is applied. A framework is also developed for obtaining exact solutions and tight lower bounds for the problem under various conditions, which include specific families of demand distributions. This is used to assess the heuristic's performance. Finally, numerical results are presented for various problem instances to illustrate the ideas