Pattern Search Ranking and Selection Algorithms for Mixed-Variable Optimization of Stochastic Systems

A new class of algorithms is introduced and analyzed for bound and linearly constrained optimization problems with stochastic objective functions and a mixture of design variable types. The generalized pattern search (GPS) class of algorithms is extended to a new problem setting in which objective function evaluations require sampling from a model of a stochastic system. The approach combines GPS with ranking and selection (R&S) statistical procedures to select new iterates. The derivative-free algorithms require only black-box simulation responses and are applicable over domains with mixed variables (continuous, discrete numeric, and discrete categorical) to include bound and linear constraints on the continuous variables. A convergence analysis for the general class of algorithms establishes almost sure convergence of an iteration subsequence to stationary points appropriately defined in the mixed-variable domain. Additionally, specific algorithm instances are implemented that provide computational enhancements to the basic algorithm. Implementation alternatives include the use modern R&S procedures designed to provide efficient sampling strategies and the use of surrogate functions that augment the search by approximating the unknown objective function with nonparametric response surfaces. In a computational evaluation, six variants of the algorithm are tested along with four competing methods on 26 standardized test problems. The numerical results validate the use of advanced implementations as a means to improve algorithm performance

Estimation of semiparametric stochastic frontiers under shape constraints with application to pollution generating technologies

A number of studies have explored the semi- and nonparametric estimation of stochastic frontier models by using kernel regression or other nonparametric smoothing techniques. In contrast to popular deterministic nonparametric estimators, these approaches do not allow one to impose any shape constraints (or regularity conditions) on the frontier function. On the other hand, as many of the previous techniques are based on the nonparametric estimation of the frontier function, the convergence rate of frontier estimators can be sensitive to the number of inputs, which is generally known as “the curse of dimensionality” problem. This paper proposes a new semiparametric approach for stochastic frontier estimation that avoids the curse of dimensionality and allows one to impose shape constraints on the frontier function. Our approach is based on the singleindex model and applies both single-index estimation techniques and shape-constrained nonparametric least squares. In addition to production frontier and technical efficiency estimation, we show how the technique can be used to estimate pollution generating technologies. The new approach is illustrated by an empirical application to the environmental adjusted performance evaluation of U.S. coal-fired electric power plants.stochastic frontier analysis (SFA), nonparametric least squares, single-index model, sliced inverse regression, monotone rank correlation estimator, environmental efficiency

Calibration of a conceptual rainfall-runoff model using simulated annealing

Simulated annealing (Kirkpatrick et al, 1983) is used to estimate the parameters of a mathematical model that predicts the water yield from a catchment. The calibration problem involves finding the global minimum of a multivariate function that has many extraneous local minima, a situation in which conventional optimisation methods are ineffective. The objective function which quantifies discrepancies between the computed and observed streamflows must be carefully selected to satisfy the least square assumptions. Several published simulated annealing algorithms have been implemented, tested and evaluated using standard test functions. Appropriate cooling schedules are found for each algorithm and test function investigated. The number of function evaluations required to find the minimum is compared to published results for the test functions using either simulated annealing and other global optimisation methods. A new simulated annealing algorithm based on the Hooke and Jeeves (1961) pattern search method is developed and compared with existing algorithms from the literature

Some applications of continuous variable neighbourhood search metaheuristic (mathematical modelling)

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.In the real world, many problems are continuous in nature. In some cases, finding the global solutions for these problems is di±cult. The reason is that the problem's objective function is non convex, nor concave and even not differentiable. Tackling these problems is often computationally too expensive. Although the development in computer technologies are increasing the speed of computations, this often is not adequate, particularly if the size of the problem's instance are large. Applying exact methods on some problems may necessitate their linearisation. Several new ideas using heuristic approaches have been considered particularly since they tackle the problems within reasonable computational time and give an approximate solution. In this thesis, the variable neighbourhood search (VNS) metaheuristic (the framework for building heuristic) has been considered. Two variants of variable neighbourhood search metaheuristic have been developed, continuous variable neighbourhood search and reformulation descent variable neighbourhood search. The GLOB-VNS software (Drazic et al., 2006) hybridises the Microsoft Visual Studio C++ solver with variable neighbourhood search metaheuristics. It has been used as a starting point for this research and then adapted and modified for problems studied in this thesis. In fact, two problems have been considered, censored quantile regression and the circle packing problem. The results of this approach for censored quantile regression outperforms other methods described in the literature, and the near-optimal solutions are obtained in short running computational time. In addition, the reformulation descent variable neighbourhood search variant in solving circle packing problems is developed and the computational results are provided

Порівняльний аналіз методів формування портфеля цінних паперів на базі копул та Марковіца

In this paper, the objects of study are securities (stocks) and portfolio.The main problem of the study is portfolio optimization. One of the first portfolio methods was presented by Henry Markowitz with his Modern Portfolio Theory (MPT), which is considered as a classic and the most popular one in modern investing. MPT provides the following assumptions: variance is used as a measure of risk, portfolio stock returns distribution is considered as a normal one. However, these assumptions do not represent real processes in the modern economy. First, in terms of modern volatile economy portfolio stock returns distribution curve has heavy tails, which is not typical for normal distribution. Secondly, in case of variance as a measure of risk probability of extreme events, such as a simultaneous increase or decrease of stock prices, are not taken into account.So Markowitz method no longer meets the requirements of the modern financial market and there is a need to study alternative and more valid portfolio methods.In this paper, copula-based approach is considered in contrast to the classical one. In the method assumption about the normality of stock returns is rejected and Value-at-Risk (VaR) is considered as a valid risk measure. VaR assessment is based on an information about random distribution. Since the normality assumption was rejected, to assess portfolio stock returns distribution need to be defined. To do these copula-functions was used.Stochastic optimization problem using VaR was solved with a modified Nelder-Mead method.As a result of the dynamic optimization return of copula-based portfolio for 2015 was 12,1 % of the initial investment sum, while the portfolio, constructed with the classical method, showed losses of 4,1 %.Since in copula-based approach incorrect normality assumption is rejected and a valid risk measure is chosen, copula-based portfolio is much more effective than the Markowitz one.В статье проведён сравнительный анализ методов формирования портфеля ценных бумаг: классического метода Марковица и метода на базе копул. Продемонстрировано, что подход на базе копул позволяет избежать некорректных предположений классического метода и более гибко описывать зависимость между случайными величинами. Результаты апробированы на данных Нью-Йоркской фондовой биржи (NYSE).У статті виконано порівняльний аналіз методів формування портфелю цінних паперів: класичного методу Марковіца та методу на базі копул. Продемонстровано, що підхід на базі копул дозволяє уникнути некоректних припущень класичного методу і більш гнучко описувати залежність між випадковими величинами. Результати апробовані на даних Нью-Йоркської фондової біржі (NYSE)