Stochastic Multi-Objective Programming Problem: A Two-Phase Weighted Coefficient Approach

This paper deals with multi-objective stochastic linear programming problem. The problem is considered by introducing the coefficients of the decision variables and the right-hand-side parameters in the constraints as normal random variables. A method for converting the problem into its deterministic problem is proposed and hence two- phase approach with equal weights is proposed for finding an efficient solution. The advantages of the approach are: as weights which is positive, not necessarily equal and generate an efficient solution. A numerical example is given to illustrate the suggested methodology.

A New Methodology for Solving Multiobjective Chance-Constrained Problems: An Application on IoT Systems

This study presents a newly developed methodology to transform the chance-constrained problem into a deterministic problem and then solving this multiobjective deterministic problem with the proposed method. Chance-constrained problem contains independent gamma random variables that are denoted as a i j . Two methods are proposed to obtain the deterministic equivalent of chance-constrained problem. The first of the methods is directly based on using the distribution, and the second consists of normalizing probabilistic constraints using Lyapunov’s central limit theorem. An algorithm which uses the Global Criterion Method is developed to solve the multiobjective deterministic equivalent of chance-constrained problem. The methodology is applied to a real-life engineering problem that consists of an IoT device and its data sending process. Using Lyapunov’s central limit theorem for large numbers of random variables is found to be more appropriate.

Optimal Stochastic Control in the Interception Problem of a Randomly Tacking Vehicle

This article considers the mathematical aspects of the problem of the optimal interception of a mobile search vehicle moving along random tacks on a given route and searching for a target, which travels parallel to this route. Interception begins when the probability of the target being detected by the search vehicle exceeds a certain threshold value. Interception was carried out by a controlled vehicle (defender) protecting the target. An analytical estimation of this detection probability is proposed. The interception problem was formulated as an optimal stochastic control problem, which was transformed to a deterministic optimization problem. As a result, the optimal control law of the defender was found, and the optimal interception time was estimated. The deterministic problem is a simplified version of the problem whose optimal solution provides a suboptimal solution to the stochastic problem. The obtained control law was compared with classic guidance methods. All the results were obtained analytically and validated with a computer simulation.

Stochastic optimization over the Pareto front by the augmented weighted Tchebychev program

В этой статье мы предлагаем новый алгоритм для решения многоцелевых задач стохастического целочисленного линейного программирования (MOSILP). Мы оптимизируем данную стохастическую линейную функцию φ по полному набору эффективных решений MOSILP, которые были преобразованы в эквивалентную детерминированную задачу с использованием неопределенных предположений, вводимых лицом, принимающим решения. Для этой цели мы применяем двухэтапный рекурсивный подход, при котором расширенная взвешенная программа Чебышева постепенно оптимизируется для создания эффективного решения, тем самым улучшая значение вспомогательной функции φ . Предлагаемый здесь подход определяет и решает последовательность целочисленных линейных программ с нарастающими ограничениями, так что на каждом этапе алгоритма генерируется новое эффективное решение. Для иллюстрации представлен числовой пример In this paper, we propose a novel algorithm to deal with multi-objective stochastic integer linear programming problems (MOSILP). Given a stochastic linear function φ , we will optimize it over the full set of efficient solutions of a MOSILP. We convert the latter into an equivalent deterministic problem using uncertain aspirations which are inputs specified by the decision maker. For this purpose, we adopt a 2-stage recourse approach where an augmented weighted Tchebychev program is progressively optimized to generate an efficient solution, the value of the utility function φ is improved to enumerate all efficient solutions. The approach proposed here defines and solves a sequence of progressively more constrained integer linear programs, so that a new efficient solution is generated at each step of the algorithm. A numerical example is presented for illustration

An Efficient Neurodynamic Approach to Fuzzy Chance-constrained Programming

In both practical applications and theoretical analysis, there are many fuzzy chance-constrained optimization problems. Currently, there is short of real-time algorithms for solving such problems. Therefore, in this paper, a continuous-time neurodynamic approach is proposed for solving a class of fuzzy chance-constrained optimization problems. Firstly, an equivalent deterministic problem with inequality constraint is discussed, and then a continuous-time neurodynamic approach is proposed. Secondly, a sufficient and necessary optimality condition of the considered optimization problem is obtained. Thirdly, the boundedness, global existence and Lyapunov stability of the state solution to the proposed approach are proved. Moreover, the convergence to the optimal solution of considered problem is studied. Finally, several experiments are provided to show the performance of proposed approach.

A guaranteed deterministic approach to superhedging: most unfavorable scenarios of market behaviour and moment problem

A guaranteed deterministic problem setting of super-replication with discrete time is considered: the aim of hedging of a contingent claim is to ensure the coverage of possible payout under the option contract for all admissible scenarios. These scenarios are given by means of a priori given compacts, that depend on the prehistory of prices: the increments of the price at each moment of time must lie in the corresponding compacts. The absence of transaction costs is assumed. The game-theoretical interpretation implies that the corresponding Bellman-Isaac equations hold, both for pure and mixed strategies. In the present paper, we propose a two-step method of solving the Bellman equation arising in the case of (game) equilibrium. In particular, the most unfavorable strategies of the `market can be found in the class of the distributions concentrated at most in n+1 point, where n is the number of risky assets.

On Dynamic Parallelization of Multilevel Monte Carlo Algorithm

MultiLevel Monte Carlo (MLMC) attracts great interest for numerical simulations of Stochastic Partial Differential Equations (SPDEs), due to its superiority over the standard Monte Carlo (MC) approach. MLMC combines in a proper manner many cheap fast simulations with few slow and expensive ones, the variance is reduced, and a significant speed up is achieved. Simulations with MC/MLMC consist of three main components: generating random fields, solving deterministic problem and reduction of the variance. Each part is subject to a different degree of parallelism. Compared to the classical MC, MLMC introduces “levels” on which the sampling is done. These levels have different computational cost, thus, efficiently utilizing the parallel resources becomes a non-trivial problem. The main focus of this paper is the parallelization of the MLMC Algorithm.

Optimal Unit Commitment Problem Considering Stochastic Wind Energy Penetration

Wind energy has attracted much attention as a clean energy resource with low running cost over the last decade,. However, due to the unpredictable nature of wind speed, the Unit Commitment (UC) problem including wind power becomes more difficult. Therefore, engineers and researchers are required to seek reliable models and techniques to plan the operation of thermal units in presence of wind farms. This paper presents a new attempt to solve the stochastic UC including wind energy sources. In order to achieve this, the problem is modeled as a chance-constrained optimization problem. Then, a method based on the here-and-now strategy is used to convert the uncertain power balance constraint into a deterministic constraint. The obtained deterministic problem is modeled using Mixed Integer Programming (MIP) on GAMS interface whereas the CEPLEX MIP solver is employed for its solution.

Oracle-based algorithms for binary two-stage robust optimization

In this work we study binary two-stage robust optimization problems with objective uncertainty. We present an algorithm to calculate efficiently lower bounds for the binary two-stage robust problem by solving alternately the underlying deterministic problem and an adversarial problem. For the deterministic problem any oracle can be used which returns an optimal solution for every possible scenario. We show that the latter lower bound can be implemented in a branch and bound procedure, where the branching is performed only over the first-stage decision variables. All results even hold for non-linear objective functions which are concave in the uncertain parameters. As an alternative solution method we apply a column-and-constraint generation algorithm to the binary two-stage robust problem with objective uncertainty. We test both algorithms on benchmark instances of the uncapacitated single-allocation hub-location problem and of the capital budgeting problem. Our results show that the branch and bound procedure outperforms the column-and-constraint generation algorithm.

New Bounds for the Stochastic Knapsack Problem

The knapsack problem is a problem in combinatorial optimization that seeks to maximize the objective function \(\sum_{i = 1}^{n} v_ix_i\) subject to the constraints \(\sum_{i = 1}^{n} w_ix_i \leq W\) and \(x_i \in \{0, 1\}\), where \(\mathbf{x}, \mathbf{v} \in \mathbb{R}^{n}\) and \(W\) are provided. We consider the stochastic variant of this problem in which \(\mathbf{v}\) remains deterministic, but \(\mathbf{x}\)is an \(n\)-dimensional vector drawn uniformly at random from \([0, 1]^{n}\). We establish a sufficient condition under which the summation-bound condition is almost surely satisfied. Furthermore, we discuss the implications of this result on the deterministic problem.