Stochastic Global Optimization Algorithm Research Articles

Meta-control of an interacting-particle algorithm for global optimization

A common issue for stochastic global optimization algorithms is how to set the parameters of the sampling distribution (e.g. temperature, mutation/cross-over rates, selection rate, etc.) so that the samplings converge to the optimum effectively and efficiently. We consider an interacting-particle algorithm and develop a meta-control methodology which analytically guides the inverse temperature parameter of the algorithm to achieve desired performance characteristics (e.g. quality of the final outcome, algorithm running time, etc.). The main aspect of our meta-control methodology is to formulate an optimal control problem where the fractional change in the inverse temperature parameter is the control variable. The objectives of the optimal control problem are set according to the desired behavior of the interacting-particle algorithm. The control problem considers particles’ average behavior, rather than treating the behavior of individual particles. The solution to the control problem provides feedback on the inverse temperature parameter of the algorithm.

Basin filling algorithm for the circular packing problem with equilibrium behavioral constraints

With the background of the satellite module layout design, the circular packing problem with equilibrium behavioral constraints is a layout optimization problem and NP-hard problem in math. For lack of a powerful optimization method, this problem is hard to solve. The energy landscape paving (ELP) method is a class of stochastic global optimization algorithms based on the Monte Carlo sampling. Based on the quasiphysical strategy and the penalty function method, the problem is converted into an unconstrained optimization problem. Here by combining the improved ELP method, the gradient method based on local search and the heuristic configuration update mechanism, a new global search algorithm, basin filling algorithm, is put forward. The numerical results show that the proposed algorithm is effective to solving the circular packing problem with equilibrium behavioral constraints, and is easy to be popularized to other layout optimization problems.

Seeker optimization algorithm: A novel stochastic search algorithm for global numerical optimization

A novel heuristic search algorithm called seeker optimization algorithm (SOA) is proposed for the real-parameter optimization. The proposed SOA is based on simulating the act of human searching. In the SOA, search direction is based on empirical gradients by evaluating the response to the position changes, while step length is based on uncertainty reasoning by using a simple fuzzy rule. The effectiveness of the SOA is evaluated by using a challenging set of typically complex functions in comparison to differential evolution (DE) and three modified particle swarm optimization (PSO) algorithms. The simulation results show that the performance of the SOA is superior or comparable to that of the other algorithms.

Combinatorial fiber-tracking of the human brain

This paper presents a novel fiber-tracking algorithm, termed combinatorial tracking, which uses stochastic process modeling and global optimization algorithm for tractography. Combinatorial tracking is a probabilistic tracking algorithm that transforms the brain's white matter into a grid in which each voxel has 26 weighted connections with adjacent voxels. We model the random walk on this graph using a Markov Chain model and suggest two approaches for fiber reconstruction. In the first approach, we find the most probable paths between two voxels with prior connectivity knowledge using a shortest path algorithm. In the second approach, the all-pairs mean first passage time (MFPT) matrix M (or hitting time as referred to in the Spectral Graph theory literature) is calculated analytically. We suggest that M can be interpreted as a global connectivity matrix and use it for fiber reconstruction. We also introduce a simulation framework that can be used to calculate specific elements of the matrix M, and show how it can be employed to select the target of a fiber in a high resolution diffusion tensor imaging (DTI) dataset. Because any source and any target voxel can be connected, combinatorial tracking permits true connectivity analysis, overcoming the limitations of conventional tracking, especially stopping criteria (e.g. low FA). We applied combinatorial tracking to a standard DTI dataset and demonstrated the reconstruction of the cortico-thalamic pathway, the pyramidal decussation, and the medial cerebellar peduncle fibers. While the DTI ellipsoid served as input for the algorithms, any diffusion imaging based orientation density function (ODF) can be used. This framework can potentially turn diffusion imaging tractography into a true connectivity measure.

Stopping and restarting strategy for stochastic sequential search in global optimization

Two common questions when one uses a stochastic global optimization algorithm, e.g., simulated annealing, are when to stop a single run of the algorithm, and whether to restart with a new run or terminate the entire algorithm. In this paper, we develop a stopping and restarting strategy that considers tradeoffs between the computational effort and the probability of obtaining the global optimum. The analysis is based on a stochastic process called Hesitant Adaptive Search with Power-Law Improvement Distribution (HASPLID). HASPLID models the behavior of stochastic optimization algorithms, and motivates an implementable framework, Dynamic Multistart Sequential Search (DMSS). We demonstrate here the practicality of DMSS by using it to govern the application of a simple local search heuristic on three test problems from the global optimization literature.

Global optimization and its applications

This special issue is devoted to selected invited papers presented at the Workshop on Global Optimization held at Imperial College London, 15–17 December 2007 (http://gow2007.ps. ic.ac.uk/). The Special Issue contains eight papers ranging from algorithms andmethodology to applications in engineering and finance. Methodological developments are considered by four papers. Floudas and Gounaris present a review of recent research in deterministic global optimization. It covers twice continuously differentiable nonlinear optimization, mixed-integer nonlinear optimization, optimization with differential-algebraic models, semi-infinite programming, optimization with grey box/nonfactorable models, and bilevel nonlinear optimization. Further methodological issues are discussed by Lasserre in relation to moments and sums of squares for polynomial optimization and related problems. The duality between moment problems and sums of squares representations of positive polynomials is central. Subsequently, the paper discusses how such results are used to define convergent semidefinite programming relaxations in polynomial optimization as well as computing the convex envelope of a rational function and finding all zeros of a system of polynomial equations. Mitsos, Chachuat, Barton consider global bilevel dynamic optimization. A deterministic algorithm for bilevel programs with nonconvex functions is given, followed by a summary of deterministic algorithms for the global solution of optimization problems with nonlinear ordinary differential equations embedded. Parpas and Rustem present a stochastic global optimization algorithm for general non-convex, smooth functions. The algorithm follows the trajectory of an appropriately defined stochastic differential equation (SDE). In order to achieve feasibility of the trajectory information from the Lagrange multipliers is introduced into the SDE. The convergence analysis is provided. Two mixed-integer linear programming (MILP)-related papers are included. The first, by McAllister, DiMaggio, Floudas, presents a computational study for solving the distance-

Convergence analysis of a global optimization algorithm using stochastic differential equations

We establish the convergence of a stochastic global optimization algorithm for general non-convex, smooth functions. The algorithm follows the trajectory of an appropriately defined stochastic differential equation (SDE). In order to achieve feasibility of the trajectory we introduce information from the Lagrange multipliers into the SDE. The analysis is performed in two steps. We first give a characterization of a probability measure (?) that is defined on the set of global minima of the problem. We then study the transition density associated with the augmented diffusion process and show that its weak limit is given by ?.

More efficient PEST compatible model independent model calibration

This article describes some of the capabilities encapsulated within the Model Independent Calibration and Uncertainty Analysis Toolbox (MICUT), which was written to support the popular PEST model independent interface. We have implemented a secant version of the Levenberg–Marquardt (LM) method that requires far fewer model calls for local search than the PEST LM methodology. Efficiency studies on three distinct environmental model structures (HSPF, FASST, and GSSHA) show that we can find comparable local minima with 36–84% fewer model calls than a conventional model independent LM application. Using the secant LM method for local search, MICUT also supports global optimization through the use of a slightly modified version of a stochastic global search technique called Multi-Level Single Linkage [Rinnooy Kan, A.H.G., Timmer, G., 1987a. Stochastic global optimization methods, part I: clustering methods. Math. Program. 39, 27–56; Rinnooy Kan, A.H.G., Timmer, G., 1987b. Stochastic global optimization methods, part ii: multi level methods. Math. Program. 39, 57–78.]. Comparison studies with three environmental models suggest that the stochastic global optimization algorithm in MICUT is at least as, and sometimes more efficient and reliable than the global optimization algorithms available in PEST.

Corrections to “A Robust Stochastic Genetic Algorithm (StGA) for Global Numerical Optimization”

In the above titled paper (ibid., vol 8, no. 5, pp. 456-470, Oct 04), there is an error in the programming of the stochastic genetic algorithm (StGA) presented in that paper. The origin and implications of the error are discussed here.

Optimally balancing energy consumption versus latency in sensor network routing

We consider wireless sensor networks with nodes switching ON (awake) and OFF (sleeping) to preserve energy, and transmitting data over channels with varying quality. The objective is to determine the best path from each node to a single gateway. The performance metrics we are interested in are: the expected energy consumption, and the probability that the latency exceeds a certain threshold. Under Markovian assumptions on the sleeping schedules and the channel conditions, we obtain the expected energy consumption of transmitting a packet on any path to the gateway. We also provide an upper (Chernoff) bound and a tight large deviations asymptotic for the latency probability on each path. To capture the trade-off between energy consumption and latency probability, we formulate the problem of choosing a path to minimize a weighted sum of the expected energy consumption and the exponent of the latency probability. We provide two algorithms to solve this problem: a centralized stochastic global optimization algorithm, and a distributed algorithm based on simulated annealing. The proposed methodology can also optimize over the fraction of time that sensor nodes remain ON (duty cycle).

Fuzzy entropy image segmentation based on particle swarm optimization

Particle swarm optimization is a stochastic global optimization algorithm that is based on swarm intelligence. Because of its excellent performance, particle swarm optimization is introduced into fuzzy entropy image segmentation to select the optimal fuzzy parameter combination and fuzzy threshold adaptively. In this study, the particles in the swarm are constructed and the swarm search strategy is proposed to meet the needs of the segmentation application. Then fuzzy entropy image segmentation based on particle swarm optimization is implemented and the proposed method obtains satisfactory results in the segmentation experiments. Compared with the exhaustive search method, particle swarm optimization can give the same optimal fuzzy parameter combination and fuzzy threshold while needing less search time in the segmentation experiments and also has good search stability in the repeated experiments. Therefore, fuzzy entropy image segmentation based on particle swarm optimization is an efficient and promising segmentation method.

A nonlinear optimization model for optimal order quantities with stochastic demand rate and price change

An inventory problem in which the stochastic demand rate in each period is considered. A model is presented to compute optimal order quantities and optimal delivery points in the planning period. This model can also account for any anticipated price change that may occur from time to time. In addition the model can be used to compute volume discounts in accordance to the size of the order. A stochastic global optimization algorithm is used to obtain the numerical results.

Multiple equilibria in a stochastic implementation of DICE with abrupt climate change

Integrated assessment modeling of global climate change has focused primarily on gradually occurring changes in the climate system. However, atmospheric and earth scientists have become increasingly concerned that the climate system may be subject to abrupt, discontinuous changes on short time scales, and that anthropogenic greenhouse-gas emissions could trigger such shifts. Incorporating this type of climate dynamics into economic or integrated assessment models can result in model non-convexity and multiple equilibria, and thus complicate policy analysis relative to models with unique, globally optimal policies. Using a version of the Nordhaus DICE model amended in previous research by Keller et al. (2004) [Keller, Klaus, Benjamin M. Bolker, David F. Bradford, 2004. Uncertain climate thresholds and optimal economic growth. Journal of environmental economics and management 48 (1), 723–741], in conjunction with a stochastic global optimization algorithm, we generate “level sets” of solutions, which helps identify multiple equilibria resulting from the potential abrupt cessation of the North Atlantic Thermohaline Circulation. We discuss the implications of this model geometry for formulating greenhouse-gas abatement policy under uncertainty and suggest that this general approach may be useful for addressing a wide range of model non-convexities including those related to endogenous technological change.

Sampling schedule design towards optimal drug monitoring for individualizing therapy

We study the individualization of therapy by simultaneously taking into account the design of sampling schedule and optimal therapeutic drug monitoring. The sampling schedule design in this work is to determine the number of samples, the sampling times, the switching time from the loading to the maintenance period, and the drug dosages. A closed-loop control policy is employed to determine the sampling schedule, and an advanced stochastic global optimization algorithm, which integrates the stochastic approximation and simulated annealing techniques, is implemented to search the optimal sampling schedule. A simulated one-compartment model of intravenous theophylline therapy is used to illustrate our method. This method can be readily extended to multiple compartment systems and allow incorporating other criteria of drug control. While currently the method is mainly of theoretical interest, it offers a starting point for practical applications and thus is hopefully of great value for the clinically individualizing therapy in the future.

A generalized framework for solving dynamic optimization problems using the artificial chemical process paradigm: Applications to particulate processes and discrete dynamic systems

The solution of optimal control problems (OCPs) becomes a challenging task when the analyzed system includes non-convex, non-differentiable, or equation-free models in the set of constraints. To solve OCPs under such conditions, a new procedure, LARES-PR, is proposed. The procedure is based on integrating the LARES algorithm with a generalized representation of the control function. LARES is a global stochastic optimization algorithm based on the artificial chemical process paradigm. The generalized representation of the control function consists of variable-length segments, which permits the use of a combination of different types of finite elements (linear, quadratic, etc.) and/or specialized functions. The functional form and corresponding parameters are determined element-wise by solving a combinatorial optimization problem. The element size is also determined as part of the solution of the optimization problem, using a novel two-step encoding strategy. These building blocks result in an algorithm that is flexible and robust in solving optimal control problems. Furthermore, implementation is very simple. The algorithm's performance is studied with a challenging set of benchmark problems. Then LARES-PR is utilized to solve optimal control problems of systems described by population balance equations, including crystallization, nano-particle formation by nucleation/coalescence mechanism, and competitive reactions in a disperse system modeled by the Monte Carlo method. The algorithm is also applied to solving the DICE model of global warming, a complex discrete-time model.

A Numerical Evaluation of Several Stochastic Algorithms on Selected Continuous Global Optimization Test Problems

There is a need for a methodology to fairly compare and present evaluation study results of stochastic global optimization algorithms. This need raises two important questions of (i) an appropriate set of benchmark test problems that the algorithms may be tested upon and (ii) a methodology to compactly and completely present the results. To address the first question, we compiled a collection of test problems, some are better known than others. Although the compilation is not exhaustive, it provides an easily accessible collection of standard test problems for continuous global optimization. Five different stochastic global optimization algorithms have been tested on these problems and a performance profile plot based on the improvement of objective function values is constructed to investigate the macroscopic behavior of the algorithms. The paper also investigates the microscopic behavior of the algorithms through quartile sequential plots, and contrasts the information gained from these two kinds of plots. The effect of the length of run is explored by using three maximum numbers of function evaluations and it is shown to significantly impact the behavior of the algorithms.

On the Investigation of Stochastic Global Optimization Algorithms

This discussion paper for the SGO 2001 Workshop considers the process of investigating stochastic global optimization algorithms. It outlines a general plan for the systematic study of their behavior. It raises questions about performance criteria, characteristics of test cases and classification of algorithms.

Genetic algorithm for constrained global optimization in continuous variables

We present a stochastic global optimization algorithm, referred to as a Genetic Algorithm (GA), for solving constrained optimization problems over a compact search domain. It is a real-coded GA that converges in probability to the optimal solution. The constraints are treated through a repair operator. A specific repair operator is included for linear inequality constraints.

On stochastic global optimization of one-dimensional functions

We consider the applicability of stochastic global optimization algorithms on test-functions whose domain of definition is a simply-connected and finite interval of real numbers. We argue on the basis of theoretical reflections of statistical physics (namely random-walk) and computer simulations that there is a decisive difference between test-problems in one and multiple dimensions pointing to the necessity to only consider test-functions in higher dimensions. We argue that only test-problems in two or more dimensions provide for the possibility to discriminate the efficiency of stochastic global optimization algorithms with respect to the complexity of the underlying physical system at all.

Implementing Pure Adaptive Search with Grover's Quantum Algorithm

Pure adaptive search (PAS) is an idealized stochastic algorithm for unconstrained global optimization. The number of PAS iterations required to solve a problem increases only linearly in the domain dimension. However, each iteration requires the generation of a random domain point uniformly distributed in the current improving region. If no regularity conditions are known to hold for the objective function, then this task requires a number of classical function evaluations varying inversely with the proportion of the domain constituted by the improving region, entirely counteracting the PAS apparent speedup. The Grover quantum computational search algorithm provides a way to generate the PAS iterates. We show that the resulting implementation, which we call the Grover adaptive search (GAS), realizes PAS for functions satisfying certain conditions, and we believe that, when quantum computers will be available, GAS will be a practical algorithm.