A Multilevel Stochastic Collocation Algorithm for Optimization of PDEs with Uncertain Coefficients

A nonlinear interval number programming method for uncertain optimization problems

ABSTRACTThe evaluation of the probabilistic constraints in reliability-based design optimization (RBDO) problems has always been significant and challenging work, which strongly affects the performance of RBDO methods. This article deals with RBDO problems using a recently developed generalized subset simulation (GSS) method and a posterior approximation approach. The posterior approximation approach is used to transform all the probabilistic constraints into ordinary constraints as in deterministic optimization. The assessment of multiple failure probabilities required by the posterior approximation approach is achieved by GSS in a single run at all supporting points, which are selected by a proper experimental design scheme combining Sobol’ sequences and Bucher’s design. Sequentially, the transformed deterministic design optimization problem can be solved by optimization algorithms, for example, the sequential quadratic programming method. Three optimization problems are used to demonstrate the efficiency and accuracy of the proposed method.

The linear quadratic optimal control problem under quadratic constraints is an optimization problem over a generally non-convex set. Yakubovich [8] and Megretski [4] have studied this problem, and they show how it may be translated into a two-stage, convex optimization problem. In this paper we study the linear quadratic control problem under quadratic constraints for generalized first-order systems. As in the state-space case the linear quadratic control problem without quadratic constraints may be solved in terms of a linear matrix inequality. Subsequently, we use the results from [8] to derive a linear matrix inequality characterizing the linear quadratic optimal behaviour under quadratic constraints.

Operation optimization of natural gas pipelines has received increasing attentions, due to such advantages as maximizing the operating economic benefit and the gas delivery amount. This paper provides a review on the most relevant research progress related to the steady-state operation optimization models of natural gas pipelines as well as corresponding solution methods based on stochastic optimization algorithms. The existing operation optimization model of the natural gas pipeline is a mixed-integer nonlinear programming (MINLP) model involving a nonconvex feasible region and mixing of continuous, discrete, and integer optimization variables, which represents an extremely difficult problem to be solved by use of optimization algorithms. A survey on the state of the art demonstrates that many stochastic algorithms show better performance of solving such optimization models due to their advantages of handling discrete variables and of high computation efficiency over classical deterministic optimization algorithms. The essential progress mainly with regard to the applications of the Genetic Algorithm (GA), Particle Swarm Optimization (PSO), Ant Colony Optimization (ACO), Simulated Annealing (SA) algorithms, and their extensions is summarized. The performances of these algorithms are compared in terms of the quality of optimization results and the computation efficiency. Furthermore, the research challenges of improving the optimization model, enhancing the stochastic algorithms, developing an online optimization technology, researching the transient optimization, and studying operation optimization of the integrated energy network are discussed.

In this chapter, we will study an important class of convex optimization problems whose objective function is given by the summation of many components. With wide applications in machine learning and distributed optimization, these problems can be viewed either as deterministic optimization problems with a special finite-sum structure or stochastic optimization problems with a discrete distribution. Accordingly, we will study two typical classes of randomized algorithms for solving them. While the first class incorporates random block decomposition into the dual space of the primal–dual methods for deterministic convex optimization, the second one employs variance reduction techniques into stochastic gradient descent methods for stochastic optimization.

Stochastic optimization algorithms — an application to pattern matching

On the stochastic linear quadratic control problem with piecewise constant admissible controls

For global shippers (i.e., multinational manufacturers or retailers), strategic sourcing of container liner shipping services from ocean carriers is an important element of successful global supply chain management. We study how to utilize information about different ocean carriers’ liner service schedules so that shippers can optimize their sourcing decisions over the selection of carriers to transport container cargoes around the world through multiple shipping lanes. We first introduce a deterministic optimization problem, where cargo demands are given in advance and in which inventory holding costs are included to capture the impacts of liner service schedules on the total operating cost. We formulate the deterministic optimization problem as a mixed integer linear program, which can be simplified by exploiting a totally unimodular property, and can thus be solved directly by a general optimization solver. To further capture the impact of liner service schedules on shippers’ resilience to the uncertainties in cargo demands, we then introduce a two‐stage robust optimization counterpart of the deterministic problem based on a probability‐free demand uncertainty set. As the two‐stage robust optimization problem is challenging to solve, we derive some novel reformulations that can enable us to develop an effective solution method. By exploiting the information about liner service schedules, we show how our proposed models and solution method can help shippers to optimize their liner service procurement decisions.

We consider minimizing a class of nonconvex composite stochastic optimization problems, and deterministic optimization problems whose objective function consists of an expectation function (or an average of finitely many smooth functions) and a weakly convex but potentially nonsmooth function. And in this paper, we focus on the theoretical properties of two types of stochastic splitting methods for solving these nonconvex optimization problems: stochastic alternating direction method of multipliers and stochastic proximal gradient descent. In particular, several inexact versions of these two types of splitting methods are studied. At each iteration, the proposed schemes inexactly solve their subproblems by using relative error criteria instead of exogenous and diminishing error rules, which allows our approaches to handle some complex regularized problems in statistics and machine learning. Under mild conditions, we obtain the convergence of the schemes and their computational complexity related to the evaluations on the component gradient of the smooth function, and find that some conclusions of their exact counterparts can be recovered.

To transfer the learned knowledge that is coming from benchmarking studies, which involve stochastic optimization algorithms, we should find a way to decide if the statistical significance between their performance is also important for real-world applications. For this reason, we have recently proposed a practical Deep Statistical Comparison (pDSC). It takes into account practical significance when making a statistical comparison of meta-heuristic stochastic optimization algorithms for single-objective optimization problems. Experimental results showed that our recently proposed approach provided very promising results.

This paper presents a genetic algorithm (GA) approach to solve the redundancy allocation problem (RAP) with imprecise parameters. The impreciseness has been considered here to tackle the unpredictable environment in real-life affairs which is directly striking the design parameters. Imprecise parameters are represented by several types of representations, viz. fuzzy, stochastic, and deterministic interval-valued. In the case of fuzzy and stochastic representations, the corresponding problems have been transformed into a deterministic interval optimization problem. Hence, to solve the problem, a GA based penalty function technique is proposed. The constraint(s) of the same are handled by constraints satisfaction rule which has been proposed in this paper. In the proposed approach, the constrained optimization problem has been converted into an unconstrained optimization problem by use of Big-M penalty technique. To solve the RAP with imprecise parameters, GA has been implemented. Finally, numerical experiments have been done for illustration purpose.

Linear Quadratic Optimal Control of a Wave Equation with Boundary Damping and Pointwise Control Input

Adaptive critic (AC) methods have common roots as generalisations of dynamic programming for neural reinforcement learning approaches. Since they approximate the dynamic programming solutions, they are potentially suitable for learning in noisy, non-linear and non-stationary environments. In this study, a novel probabilistic dual heuristic programming (DHP)-based AC controller is proposed. Distinct to current approaches, the proposed probabilistic (DHP) AC method takes uncertainties of forward model and inverse controller into consideration. Therefore, it is suitable for deterministic and stochastic control problems characterised by functional uncertainty. Theoretical development of the proposed method is validated by analytically evaluating the correct value of the cost function which satisfies the Bellman equation in a linear quadratic control problem. The target value of the probabilistic critic network is then calculated and shown to be equal to the analytically derived correct value. Full derivation of the Riccati solution for this non-standard stochastic linear quadratic control problem is also provided. Moreover, the performance of the proposed probabilistic controller is demonstrated on linear and non-linear control examples.

In this paper, a new stochastic optimization algorithm is introduced to simulate the plant growing process. It employs the photosynthesis operator and phototropism operator to mimic photosynthesis and phototropism phenomenon. For the plant growing process, photosynthesis is a basic mechanism to provide the energy from sunshine, while phototropism is an important character to guide the growing direction. In our algorithm, each individual is called a branch, and the sampled points are regarded as the branch growing trajectory. Phototropism operator is designed to introduce the fitness function value, as well as phototropism operator is used to decide the growing direction. To test the performance, it is used to produce initial values for constrained optimization problems. Simulation results show this new algorithm increases the performance significantly when compared with method of randomly generating.

In this paper, we present a power control algorithm in cognitive radio multiple-input–multiple-output (MIMO) system over time-varying fading channels based on the utility-based framework. In particular, a more general objective function framework is proposed under a long- or short-term transmitting and interference power constraints for secondary users, which results the optimization problem is nonconvex. A stochastic optimization algorithm is proposed based on an improved quantum-behaved particle swarm optimization (IQPSO), which can solve the nonconvex optimization problem efficiently. The acceleration coefficients are adjusted with time-varying equations which can help the particles jump out the local optimum. Due to the characteristics of wave function of quantum Delta potential well model, IQPSO can get better optimal solutions in search space for the stochastic optimization problem. To show the efficiency of the proposed algorithm, the utilities of the MIMO CR system got by IQPSO are compared with other approaches in the literature in different case studies, which show that the proposed algorithm can solve the nonconvex problem efficiently and achieve significant throughput.