Stochastic optimization algorithms for barrier dividend strategies

This work focuses on finding optimal barrier policy for an insurance risk model when the dividends are paid to the share holders according to a barrier strategy. A new approach based on stochastic optimization methods is developed. Compared with the existing results in the literature, more general surplus processes are considered. Precise models of the surplus need not be known; only noise-corrupted observations of the dividends are used. Using barrier-type strategies, a class of stochastic optimization algorithms are developed. Convergence of the algorithm is analyzed; rate of convergence is also provided. Numerical results are reported to demonstrate the performance of the algorithm.     

Stochastic variance-reduced prox-linear algorithms for nonconvex composite optimization

Mathematical Programming - We consider the problem of minimizing composite functions of the form $$f(g(x))+h(x)$$ , where f and h are convex functions (which can be nonsmooth) and...     

A mean field model for a class of garbage collection algorithms in flash-based solid state drives

Garbage collection (GC) algorithms play a key role in reducing the write amplification in flash-based solid state drives, where the write amplification affects the lifespan and speed of the drive. This paper introduces a mean field model to assess the write amplification and the distribution of the number of valid pages per block for a class $\mathcal {C}$ of GC algorithms. Apart from the Random GC algorithm, class $\mathcal {C}$ includes two novel GC algorithms: the $d$ -Choices GC algorithm, that selects $d$ blocks uniformly at random and erases the block containing the least number of valid pages among the $d$ selected blocks, and the Random++ GC algorithm, that repeatedly selects another block uniformly at random until it finds a block with a lower than average number of valid blocks. Using simulation experiments, we show that the proposed mean field model is highly accurate in predicting the write amplification (for drives with $N=50{,}000$ blocks). We further show that the $d$ -Choices GC algorithm has a write amplification close to that of the Greedy GC algorithm even for small $d$ values, e.g., $d = 10$ , and offers a more attractive trade-off between its simplicity and its performance than the Windowed GC algorithm introduced and analyzed in earlier studies. The Random++ algorithm is shown to be less effective as it is even inferior to the FIFO algorithm when the number of pages $b$ per block is large (e.g., for $b \ge 64$ ).     

Stochastic modeling of unresolved scales in complex systems

Model uncertainties or simulation uncertainties occur in mathematical modeling of multiscale complex systems, since some mechanisms or scales are not represented (i.e., ‘unresolved’) due to a lack in our understanding of these mechanisms or limitations in computational power. The impact of these unresolved scales on the resolved scales needs to be parameterized or taken into account. A stochastic scheme is devised to take the effects of unresolved scales into account, in the context of solving nonlinear partial differential equations. An example is presented to demonstrate this strategy. Dedicated to Professor Peter E. Kloeden on the occasion of his 60th birthday     

Stochastic modelling and optimization of water resources systems

This work involves the simultaneous optimization of the initial design and operating policy over the life of multipurpose multireservoir water resources systems receiving stochastic inflows. The approach is based on the division of the reservoir into two imaginary water storage pools, namely, the conservation and flood pools. Based on this treatment, the optimization problem is stated using the concepts of Lagrange multipliers and parameter optimization. Two nonlinear programming techniques, namely, the generalized reduced gradient technique and the gradient projection technique, combined independently with Markovian decision are proposed to solve such a problem. To illustrate the use of the proposed techniques, the Walnut River Basin in southeastern Kansas, is employed in this work.     

Stochastic service systems,random interval graphs and search algorithms

We consider several stochastic service systems, and study the asymptotic behavior of the moments of various quantities that have application to models for random interval graphs and algorithms for searching for an idle server or for an vacant or occupied waiting station. In some cases the moments turn out to involve Lambert series for the generating functions for the sums of powers of divisors of positive integers. For these cases we are able to obtain complete asymptotic expansions for the moments of the quantities in question. © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 45, 421–442, 2014     

Stochastic design optimization of asynchronous flexible assembly systems

This paper presents the application of the stochastic quasigradient method (SQG) of Ermoliev and Gaivaronski to the performance optimization of asynchronous flexible assembly systems (AFAS). These systems are subject to blocking and starvation effects that make complete analytic performance modeling difficult. A hybrid algorithm is presented in this paper which uses a queueing network model to set the number of pallets in the system and then an SQG algorithm is used to set the buffer spacings to obtain optimal system throughput. Different forms of the SQG algorithm are examined and the specification of optimal buffer sizes and pallet numbers for a variety of assembly systems with ten stations are discussed. The combined Network-SQG method appears to perform well in obtaining a near optimal solution in this discrete optimization example, even though the SQG method was primarily designed for application to differentiable performance functionals. While a number of both theoretical and practical problems remain to be resolved, a heuristic version of the SQG method appears to be a reasonable technique for analyzing optimization problems for certain complex manufacturing systems.     

Optimal memory management in a system with garbage collection

This paper discusses the problem of determining cost-saving policies for obtaining memory in systems with garbage collection, in particular the run time system of DEC-10 Simula. A mathematical model of the problem is defined and a policy is obtained from the model and a certain amount of heuristic reasoning. The implementation and validation of the policy is also described.     

Stochastic modeling of quality of systems operating in a heterogeneous environment

We consider systems that are subject to an external mixed Poisson shock process. Each shock can result in a failure of a system with a given probability and is survived with the complementary probability. Each shock additionally decreases the quality function that describes the performance of a system, thus forming the corresponding stochastic process. Expectations (unconditional and conditional on survival) and relevant variability characteristics for the stochastic quality function are derived. Some monotonicity properties of the conditional quality function are investigated and the future values of this function are derived.     

Stochastic quasigradient methods for optimization of discrete event systems

In this paper, stochastic programming techniques are adapted and further developed for applications to discrete event systems. We consider cases where the sample path of the system depends discontinuously on control parameters (e.g. modeling of failures, several competing processes), which could make the computation of estimates of the gradient difficult. Methods which use only samples of the performance criterion are developed, in particular finite differences with reduced variance and concurrent approximation and optimization algorithms. Optimization of the stationary behavior is also considered. Results of numerical experiments and convergence results are reported.     

Local and global optimization by parallel algorithms for MIMD systems

This paper describes an implementation on the Neptune system at Loughborough University of Sutti's parallel (MIMD) algorithm [1–3] and an analysis of its performance. Parallel asynchronous versions of Powell's method [6] and Price's algorithm [7] are proposed, designed for efficient implementation on MIMD systems.This work has been developed during the author's stay at the Numerical Optimization Centre, Hatfield Polytechnic, England.     

Optimal control of bioprocess systems using hybrid numerical optimization algorithms

In the paper, we consider the bioprocess system optimal control problem. Generally speaking, it is very difficult to solve this problem analytically. To obtain the numerical solution, the problem is transformed into a parameter optimization problem with some variable bounds, which can be efficiently solved using any conventional optimization algorithms, e.g. the improved Broyden–Fletcher–Goldfarb–Shanno algorithm. However, in spite of the improved Broyden–Fletcher–Goldfarb–Shanno algorithm is very efficient for local search, the solution obtained is usually a local extremum for non-convex optimal control problems. In order to escape from the local extremum, we develop a novel stochastic search method. By performing a large amount of numerical experiments, we find that the novel stochastic search method is excellent in exploration, while bad in exploitation. In order to improve the exploitation, we propose a hybrid numerical optimization algorithm to solve the problem based on the novel stochastic search method and the improved Broyden–Fletcher–Goldfarb–Shanno algorithm. Convergence results indicate that any global optimal solution of the approximate problem is also a global optimal solution of the original problem. Finally, two bioprocess system optimal control problems illustrate that the hybrid numerical optimization algorithm proposed by us is low time-consuming and obtains a better cost function value than the existing approaches.     

First-order optimization algorithms via inertial systems with Hessian driven damping

For controlled discrete-time stochastic processes we introduce a new class of dynamic risk measures, which we call process-based. Their main feature is that they measure risk of processes that are functions of the history of a base process. We introduce a new concept of conditional stochastic time consistency and we derive the structure of process-based risk measures enjoying this property. We show that they can be equivalently represented by a collection of static law-invariant risk measures on the space of functions of the state of the base process. We apply this result to controlled Markov processes and we derive dynamic programming equations. We also derive dynamic programming equations for multistage stochastic programming with decision-dependent distributions.

     

Multiphase flow modeling and optimization for online wash systems of gas turbines

Compressor blade fouling in gas turbines could lead to significant loss in power and efficiency. Online wash systems are used to remove fouling without having to shut down the gas turbine. Hence, the design of efficient online wash systems could lead to significant recovery of performance. The present study aims at developing a generalized numerical approach for the optimization of online wash systems for gas turbines. The procedure utilizes a Lagrangian particle-tracking multiphase flow model combined with constrained multi-objective optimization to develop design charts for selecting the optimal nozzle diameter, location, mass flow rate and half-cone angle. The developed approach is used to optimize the design of the online wash system for the General Electric MS5002 gas turbine model. Comparisons and differences between the optimized design and the existing design are highlighted and presented.     

Genetic algorithms in constrained optimization

The behavior of the two-point crossover operator, on candidate solutions to an optimization problem that is restricted to integer values and by some set of constraints, is investigated theoretically. This leads to the development of new genetic operators for the case in which the constraint system is linear.The computational difficulty asserted by many optimization problems has lead to exploration of a class of randomized algorithms based on biological adaption. The considerable interest that surrounds these evolutionary algorithms is largely centered on problems that have defied satisfactory illation by traditional means because of badly behaved or noisy objective functions, high dimensionality, or intractable algorithmic complexity. Under such conditions, these alternative methods have often proved invaluable.Despite their attraction, the applicability of evolutionary algorithms has been limited by a deficiency of general techniques to manage constraints, and the difficulty is compounded when the decision variables are discrete. Several new genetic operators are presented here that are guaranteed to preserve the feasibility of discrete aspirant solutions with respect to a system of linear constraints.To avoid performance degradation as the probability of finding a feasible and meaningful information exchange between two candidate solutions decreases, relaxations of the modified genetic crossover operator are also proposed. The effective utilization of these also suggests a manipulation of the genetic algorithm itself, in which the population is evanescently permitted to grow beyond its normal size.     

Stochastic modeling of phytoplankton allelopathy

The study of dynamic interactions between two competing phytoplankton species in the presence of toxic substances is an active field of research due to the global increase of harmful phytoplankton blooms. Ordinary differential equation models for two competing phytoplankton species, when one or both the species liberate toxic substances, are unable to capture the oscillatory and highly variable growth of phytoplankton populations. The deterministic formulation never predicts the sudden localized extinction of certain species. These obstacles of mathematical modeling can be overcome if we include stochastic variability in our modeling approach. In this investigation, we construct stochastic models of allelopathic interactions between two competing phytoplankton species as a continuous time Markov chain model as well as an Itô stochastic differential equation model. Approximate extinction probabilities for both species are obtained analytically for the continuous time Markov chain model. Analytical estimates are validated with the help of numerical simulations.     

Connectivity modeling and optimization of linear consecutively connected systems with repairable connecting elements

Linear consecutively connected systems (LCCSs) are systems containing a linear sequence of ordered nodes. Connection elements (CE) characterized by diverse connection ranges, time-to-failure and time-to-repair distributions are allocated to different nodes to provide the system connectivity, i.e., a connection between the source and sink nodes of the LCCS. Examples of LCCSs abound in practical applications such as flow transmission systems and radio communication systems. Considerable research efforts have been expended in modeling and optimizing LCCSs. However, most of the existing works have assumed that CEs either are non-repairable or undergo a restrictive minimal repair policy with constant repair time. This paper makes new technical contributions by modeling and optimizing LCCSs with CEs under corrective maintenance with random repair time and different repair policies (minimal, perfect, and imperfect). The characteristics of CEs can depend on their location because the distance between adjacent nodes and conditions of CE operation and maintenance at different nodes can be different, which further complicates the problem. We first propose a discrete numerical algorithm to evaluate the instantaneous availability of each CE. A universal generating function based method is then implemented for assessing instantaneous and expected system connectivity for a specific CE allocation. As the CE allocation can have significant impacts on the system connectivity, we further define and solve the optimal CE allocation problem, whose objective is to find the CE allocation among LCCS nodes maximizing the expected system connectivity over a given mission time. Effects of different parameters including repair efficiency, mission time and repair time are investigated. As illustrated through examples, optimization results can facilitate optimal decisions on robust design and effective operation and maintenance managements of LCCSs.     

On the use of two QMR algorithms for solving singular systems and applications in Markov chain modeling

Recently, Freund and Nachtigal proposed the quasi-minimal residual algorithm (QMR) for solving general nonsingular non-Hermitian linear systems. The method is based on the Lanczos process, and thus it involves matrix—vector products with both the coefficient matrix of the linear system and its transpose. Freund developed a variant of QMR, the transpose-free QMR algorithm (TFQMR), that only requires products with the coefficient matrix. In this paper, the use of QMR and TFQMR for solving singular systems is explored. First, a convergence result for the general class of Krylov-subspace methods applied to singular systems is presented. Then, it is shown that QMR and TFQMR both converge for consistent singular linear systems with coefficient matrices of index 1. Singular systems of this type arise in Markov chain modeling. For this particular application, numerical experiments are reported.     

Constraint decomposition algorithms in global optimization

Many global optimization problems can be formulated in the form min{c(x, y): x X, y Y, (x, y) Z, y G} where X, Y are polytopes in ^p , ⁿ , respectively, Z is a closed convex set in ^p+n, while G is the complement of an open convex set in ⁿ . The function c: ^p+n is assumed to be linear. Using the fact that the nonconvex constraints depend only upon they-variables, we modify and combine basic global optimization techniques such that some new decomposition methods result which involve global optimization procedures only in ⁿ . Computational experiments show that the resulting algorithms work well for problems with smalln.     

Randomized algorithms in interval global optimization

This paper is a critical survey of the interval optimization methods aimed at computing global optima for multivariable functions. To overcome some drawbacks of traditional deterministic interval techniques, we outline some ways of constructing stochastic (randomized) algorithms in interval global optimization, in particular, those based on the ideas of random search and simulated annealing.