Expected hitting times for Backtracking Adaptive Search

This article analyses a counting process associated with a stochastic process arising in global optimisation. Backtracking adaptive search (BAS) is a theoretical stochastic global optimisation algorithm modelling the temporary acceptance of solutions of lower quality. BAS generalises the pure adaptive search and hesitant adaptive search algorithms, whose full search duration distributions are known. This article gives the exact expected search duration for backtracking adaptive search.     

Hesitant adaptive search: the distribution of the number of iterations to convergence

Hesitant adaptive search is a stochastic optimisation procedure which accommodates hesitation, or pausing, at objective function values. It lies between pure adaptive search (which strictly improves at each iteration) and simulated annealing with constant temperature (which allows backtracking, or the acceptance of worse function values). In this paper we build on an earlier work and make two contributions; first, we link hesitant adaptive search to standard counting process theory, and second, we use this to derive the exact distribution of the number of iterations of hesitant adaptive search to termination. Received: November 17, 1997 / Accepted: July 9, 1999?Published online December 15, 2000     

Expected search duration for finite backtracking adaptive search

Backtracking adaptive search is an optimisation algorithm which generalises pure adaptive search and hesitant adaptive search. This paper considers the number of iterations for which the algorithm runs, on a problem with finitely many range levels, in order to reach a sufficiently extreme objective function level. A difference equation for the expectation of this quantity is derived and solved. Several examples of backtracking adaptive search on finite problems are presented, including special cases that have received attention in earlier papers.     

Generating functions and the performance of backtracking adaptive search

Backtracking adaptive search is a simplified stochastic optimisation procedure which permits the acceptance of worsening objective function values. Key properties of backtracking adaptive search are defined and obtained using generating functions. Examples are given to illustrate the use of this methodology.      

Hesitant adaptive search for global optimisation

Pure adaptive search is a stochastic algorithm which has been analysed in distinct ways for finite and continuous global optimisation. In this paper, motivated by the behaviour of practical algorithms such as simulated annealing, we extend these ideas. We present a unified theory which yields both the finite and continuous results for pure adaptive search. At the same time, we allow our extended algorithm to hesitate before improvement continues. Results are obtained for the expected number of iterations to convergence for such an algorithm. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.Corresponding author.     

Convergence guarantees for generalized adaptive stochastic search methods for continuous global optimization

This paper presents some simple technical conditions that guarantee the convergence of a general class of adaptive stochastic global optimization algorithms. By imposing some conditions on the probability distributions that generate the iterates, these stochastic algorithms can be shown to converge to the global optimum in a probabilistic sense. These results also apply to global optimization algorithms that combine local and global stochastic search strategies and also those algorithms that combine deterministic and stochastic search strategies. This makes the results applicable to a wide range of global optimization algorithms that are useful in practice. Moreover, this paper provides convergence conditions involving the conditional densities of the random vector iterates that are easy to verify in practice. It also provides some convergence conditions in the special case when the iterates are generated by elliptical distributions such as the multivariate Normal and Cauchy distributions. These results are then used to prove the convergence of some practical stochastic global optimization algorithms, including an evolutionary programming algorithm. In addition, this paper introduces the notion of a stochastic algorithm being probabilistically dense in the domain of the function and shows that, under simple assumptions, this is equivalent to seeing any point in the domain with probability 1. This, in turn, is equivalent to almost sure convergence to the global minimum. Finally, some simple results on convergence rates are also proved.     

A penalty function-based random search algorithm for optimal control of switched systems with stochastic constraints and its application in automobile test-driving with gear shifts

Practical industrial process is usually a dynamic process including uncertainty. Stochastic constraints can be used for industrial process modeling, when system sate and/or control input constraints cannot be strictly satisfied. Thus, optimal control of switched systems with stochastic constraints can be available to address practical industrial process problems with different modes. In general, obtaining an analytical solution of the optimal control problem is usually very difficult due to the discrete nature of the switching law and the complexity of stochastic constraints. To obtain a numerical solution, this problem is formulated as a constrained nonlinear parameter selection problem (CNPSP) based on a relaxation transformation (RT) technique, an adaptive sample approximation (ASA) method, a smooth approximation (SA) technique, and a control parameterization (CP) method. Following that, a penalty function-based random search (PFRS) algorithm is designed for solving the CNPSP based on a novel search rule-based penalty function (NSRPF) method and a novel random search (NRS) algorithm. The convergence results show that the proposed method is globally convergent. Finally, an optimal control problem in automobile test-driving with gear shifts (ATGS) is further extended to illustrate the effectiveness of the proposed method by taking into account some stochastic constraints. Numerical results show that compared with other typical methods, the proposed method is less conservative and can obtain a stable and robust performance when considering the small perturbations in initial system state. In addition, to balance the computation amount and the numerical solution accuracy, a tolerance setting method is also provided by the numerical analysis technique.     

Adaptive Monte Carlo Variance Reduction for Lévy Processes with Two-Time-Scale Stochastic Approximation

We propose an approach to a twofold optimal parameter search for a combined variance reduction technique of the control variates and the important sampling in a suitable pure-jump Lévy process framework. The parameter search procedure is based on the two-time-scale stochastic approximation algorithm with equilibrated control variates component and with quasi-static importance sampling one. We prove the almost sure convergence of the algorithm to a unique optimum. The parameter search algorithm is further embedded in adaptive Monte Carlo simulations in the case of the gamma distribution and process. Numerical examples of the CDO tranche pricing with the Gamma copula model and the intensity Gamma model are provided to illustrate the effectiveness of our method.      

Speeding up continuous GRASP

Continuous GRASP (C-GRASP) is a stochastic local search metaheuristic for finding cost-efficient solutions to continuous global optimization problems subject to box constraints (Hirsch et al., 2007). Like a greedy randomized adaptive search procedure (GRASP), a C-GRASP is a multi-start procedure where a starting solution for local improvement is constructed in a greedy randomized fashion. In this paper, we describe several improvements that speed up the original C-GRASP and make it more robust. We compare the new C-GRASP with the original version as well as with other algorithms from the recent literature on a set of benchmark multimodal test functions whose global minima are known. Hart’s sequential stopping rule (1998) is implemented and C-GRASP is shown to converge on all test problems.     

Adaptive brainstorm optimisation with multiple strategies

Brainstorm optimisation (BSO) algorithm is a recently developed swarm intelligence algorithm inspired by the human problem-solving process. BSO has been shown to be an efficient method for creating better ideas to deal with complex problems. The original BSO suffers from low convergence and is easily trapped in local optima due to the improper balance between global exploration and local exploitation. Motivated by the memetic framework, an adaptive BSO with two complementary strategies (AMBSO) is proposed in this study. In AMBSO, a differential-based mutation technique is designed for global exploration improvement and a sub-gradient strategy is integrated for local exploitation enhancement. To dynamically trigger the appropriate strategy, an adaptive selection mechanism based on historical effectiveness is developed. The proposed algorithm is tested on 30 benchmark functions with various properties, such as unimodal, multimodal, shifted and rotated problems, in dimensions of 10, 30 and 50 to verify their scalable performance. Six state-of-the-art optimisation algorithms are included for comparison. Experimental results indicate the effectiveness of AMBSO in terms of solution quality and convergence speed.     

SPT: a stochastic tunneling algorithm for global optimization

A stochastic approach to solving unconstrained continuous-function global optimization problems is presented. It builds on the tunneling approach to deterministic optimization presented by Barhen and co-workers (Bahren and Protopopescu, in: State of the Art in Global Optimization, Kluwer, 1996; Barhen et al., Floudas and Pardalos (eds.), TRUST: a deterministic algorithm for global optimization, 1997) by combining a series of local descents with stochastic searches. The method uses a rejection-based stochastic procedure to locate new local minima descent regions and a fixed Lipschitz-like constant to reject unpromising regions in the search space, thereby increasing the efficiency of the tunneling process. The algorithm is easily implemented in low-dimensional problems and scales easily to large problems. It is less effective without further heuristics in these latter cases, however. Several improvements to the basic algorithm which make use of approximate estimates of the algorithms parameters for implementation in high-dimensional problems are also discussed. Benchmark results are presented, which show that the algorithm is competitive with the best previously reported global optimization techniques. A successful application of the approach to a large-scale seismology problem of substantial computational complexity using a low-dimensional approximation scheme is also reported.     

Adaptive discretization of convex multistage stochastic programs

We propose a new scenario tree reduction algorithm for multistage stochastic programs, which integrates the reduction of a scenario tree into the solution process of the stochastic program. This allows to construct a scenario tree that is highly adapted on the optimization problem. The algorithm starts with a rough approximation of the original tree and locally refines this approximation as long as necessary. Promising numerical results for scenario tree reductions in the settings of portfolio management and power management with uncertain load are presented.     

An affine scaling trust-region algorithm with interior backtracking technique for solving bound-constrained nonlinear systems

In this paper, we propose a new affine scaling trust-region algorithm in association with nonmonotonic interior backtracking line search technique for solving nonlinear equality systems subject to bounds on variables. The trust-region subproblem is defined by minimizing a squared Euclidean norm of linear model adding the augmented quadratic affine scaling term subject only to an ellipsoidal constraint. By using both trust-region strategy and interior backtracking line search technique, each iterate switches to backtracking step generated by the general trust-region subproblem and satisfies strict interior point feasibility by line search backtracking technique. The global convergence and fast local convergence rate of the proposed algorithm are established under some reasonable conditions. A nonmonotonic criterion should bring about speeding up the convergence progress in some ill-conditioned cases. The results of numerical experiments are reported to show the effectiveness of the proposed algorithm.     

Random backtracking in backtrack search algorithms for satisfiability

This paper proposes the utilization of randomized backtracking within complete backtrack search algorithms for propositional satisfiability (SAT). In recent years, randomization has become pervasive in SAT algorithms. Incomplete algorithms for SAT, for example the ones based on local search, often resort to randomization. Complete algorithms also resort to randomization. These include state-of-the-art backtrack search SAT algorithms that often randomize variable selection heuristics. Moreover, it is plain that the introduction of randomization in other components of backtrack search SAT algorithms can potentially yield new competitive search strategies. As a result, we propose a stochastic backtrack search algorithm for SAT, that randomizes both the variable selection and the backtrack steps of the algorithm. In addition, we relate randomized backtracking with a more general form of backtracking, referred to as unrestricted backtracking. Finally, experimental results for different organizations of randomized backtracking are described and compared, providing empirical evidence that the new search algorithm for SAT is a very competitive approach for solving hard real-world instances.     

Adaptive nonmonotone line search method for unconstrained optimization

In this paper, an adaptive nonmonotone line search method for unconstrained minimization problems is proposed. At every iteration, the new algorithm selects only one of the two directions: a Newton-type direction and a negative curvature direction, to perform the line search. The nonmonotone technique is included in the backtracking line search when the Newton-type direction is the search direction. Furthermore, if the negative curvature direction is the search direction, we increase the steplength under certain conditions. The global convergence to a stationary point with second-order optimality conditions is established. Some numerical results which show the efficiency of the new algorithm are reported.      

Backtracking Adaptive Search: Distribution of Number of Iterations to Convergence

Backtracking adaptive search is a simplified stochastic optimiza-tion procedure which permits the acceptance of worsening objective function values. It generalizes the hesitant adaptive search, which in turn is a gener-alization of the pure adaptive search. In this paper, we use ideas from the theory of stochastic processes to determine the full distribution of the number of iterations to convergence for the backtracking adaptive search.Communicated by P. M. PardalosThe authors thank theMarsden Fund of the Royal Society of New Zealand for support of this research. The fourth author was supported by a Bright Future Scholarship administered by the Foundation for Research, Science and Technology.     

Descent Direction Stochastic Approximation Algorithm with Adaptive Step Sizes

A stochastic approximation (SA) algorithm with new adaptive step sizes for solving unconstrained minimization problems in noisy environment is proposed. New adaptive step size scheme uses ordered statistics of fixed number of previous noisy function values as a criterion for accepting good and rejecting bad steps. The scheme allows the algorithm to move in bigger steps and avoid steps proportional to $1/k$ when it is expected that larger steps will improve the performance. An algorithm with the new adaptive scheme is defined for a general descent direction. The almost sure convergence is established. The performance of new algorithm is tested on a set of standard test problems and compared with relevant algorithms. Numerical results support theoretical expectations and verify efficiency of the algorithm regardless of chosen search direction and noise level. Numerical results on problems arising in machine learning are also presented. Linear regression problem is considered using real data set. The results suggest that the proposed algorithm shows promise.     

线性不等式约束的广义非线性互补问题的仿射内点信赖域方法

提供了一种新的非单调内点回代线搜索技术的仿射内点信赖域方法解线性不等式约束的广义非线性互补问题(GCP).基于广义互补问题构成的半光滑方程组的广义Jacobian矩阵,算法使用l2范数作为半光滑方程组的势函数,形成的信赖域子问题为一个带椭球约束的线性化的二次模型.利用广义牛顿方程计算试探迭代步,通过内点映射回代技术确保迭代点是严格内点,保证了算法的整体收敛性.在合理的条件下,证明了信赖域算法在接近最优点时可转化为广义拟牛顿步,进而具有局部超线性收敛速率.非单调技术将克服高度非线性情况加速收敛进展.最后,数值结果表明了算法的有效性.     

The Million-Variable “March” for Stochastic Combinatorial Optimization

Combinatorial optimization problems have applications in a variety of sciences and engineering. In the presence of data uncertainty, these problems lead to stochastic combinatorial optimization problems which result in very large scale combinatorial optimization problems. In this paper, we report on the solution of some of the largest stochastic combinatorial optimization problems consisting of over a million binary variables. While the methodology is quite general, the specific application with which we conduct our experiments arises in stochastic server location problems. The main observation is that stochastic combinatorial optimization problems are comprised of loosely coupled subsystems. By taking advantage of the loosely coupled structure, we show that decomposition-coordination methods provide highly effective algorithms, and surpass the scalability of even the most efficiently implemented backtracking search algorithms.     

Bayesian variable selection in generalized linear models using a combination of stochastic optimization methods

In this paper the usage of a stochastic optimization algorithm as a model search tool is proposed for the Bayesian variable selection problem in generalized linear models. Combining aspects of three well known stochastic optimization algorithms, namely, simulated annealing, genetic algorithm and tabu search, a powerful model search algorithm is produced. After choosing suitable priors, the posterior model probability is used as a criterion function for the algorithm; in cases when it is not analytically tractable Laplace approximation is used. The proposed algorithm is illustrated on normal linear and logistic regression models, for simulated and real-life examples, and it is shown that, with a very low computational cost, it achieves improved performance when compared with popular MCMC algorithms, such as the MCMC model composition, as well as with “vanilla” versions of simulated annealing, genetic algorithm and tabu search.