Evaluating the quality of online optimization algorithms by discrete event simulation

A key feature of dynamic problems which offer degrees of freedom to the decision maker is the necessity for a goal-oriented decision making routine which is employed every time the logic of the system requires a decision. In this paper, we look at optimization procedures which appear as subroutines in dynamic problems and show how discrete event simulation can be used to assess the quality of algorithms: after establishing a general link between online optimization and discrete event systems, we address performance measurement in dynamic settings and derive a corresponding tool kit. We then analyze several control strategies using the methodologies discussed previously in two real world examples of discrete event simulation models: a manual order picking system and a pickup and delivery service.     

Confidence sets for discrete stochastic optimization

Consider the problem of finding the points of maximum of an expectation functional over a finite setS. Based on statistical estimates at each point ofS, confidence sets for theargmax-set are constructed which guarantee a prespecified probability of correct selection. We review known selection methods and propose a new two-stage procedure that works well for largeS and few global maxima. The performance is compared in a simulation study.     

Communication-efficient algorithms for decentralized and stochastic optimization

Mathematical Programming - We present a new class of decentralized first-order methods for nonsmooth and stochastic optimization problems defined over multiagent networks. Considering that...     

Shadow price of information in discrete time stochastic optimization

Mathematical Programming - The shadow price of information has played a central role in stochastic optimization ever since its introduction by Rockafellar and Wets in the mid-seventies. This...     

A modification of the stochastic ruler method for discrete stochastic optimization

We propose a modified stochastic ruler method for finding a global optimal solution to a discrete optimization problem in which the objective function cannot be evaluated analytically but has to be estimated or measured. Our method generates a Markov chain sequence taking values in the feasible set of the underlying discrete optimization problem; it uses the number of visits this sequence makes to the different states to estimate the optimal solution. We show that our method is guaranteed to converge almost surely (a.s.) to the set of global optimal solutions. Then, we show how our method can be used for solving discrete optimization problems where the objective function values are estimated using either transient or steady-state simulation. Finally, we provide some numerical results to check the validity of our method and compare its performance with that of the original stochastic ruler method.     

A new approach to discrete stochastic optimization problems

Solving a discrete stochastic optimization problem involves two efforts: one is to sample and search the design space; and the other is to estimate the performance values of the sampled designs. When the amount of computational resources is limited, we need to know how to balance these two efforts in order to obtain the best result. In this paper, two performance measures which quantify the marginal contribution of the searching process as well as the performance evaluation process are proposed. Using these two measures, we recommend a framework that can dynamically allocate the computational resources to the search process and the performance evaluation process. Two algorithms based on the proposed framework are tested on several scenarios, and the results produced are very promising.     

On a multistage discrete stochastic optimization problem with stochastic constraints and nested sampling

Mathematical Programming -     

Optimal strategies for asset allocation and consumption under stochastic volatility

Selecting optimal asset allocation and consumption strategies is an important, but difficult, topic in modern finance. The dynamics is governed by a nonlinear partial differential equation. Stochastic volatility adds further complication. Even to obtain a numerical solution is challenging. Here, we develop a closed-form approximate solution. We show that our theoretical predictions for the optimal asset allocation strategy and the optimal consumption strategy are in surprisingly good agreement with the results from full numerical computations.     

Single observation adaptive search for discrete and continuous stochastic optimization

Solving a stochastic optimization problem often involves performing repeated noisy function evaluations at points encountered during the algorithm. Recently, a continuous optimization framework for executing a single observation per search point was shown to exhibit a martingale property so that associated estimation errors are guaranteed to converge to zero. We generalize this martingale single observation approach to problems with mixed discrete–continuous variables. We establish mild regularity conditions for this class of algorithms to converge to a global optimum.     

A comparative study of decomposition algorithms for stochastic combinatorial optimization

This paper presents comparative computational results using three decomposition algorithms on a battery of instances drawn from two different applications. In order to preserve the commonalities among the algorithms in our experiments, we have designed a testbed which is used to study instances arising in server location under uncertainty and strategic supply chain planning under uncertainty. Insights related to alternative implementation issues leading to more efficient implementations, benchmarks for serial processing, and scalability of the methods are also presented. The computational experience demonstrates the promising potential of the disjunctive decomposition (D ²) approach towards solving several large-scale problem instances from the two application areas. Furthermore, the study shows that convergence of the D ² methods for stochastic combinatorial optimization (SCO) is in fact attainable since the methods scale well with the number of scenarios.     

Optimal estimation of a state based on deterministic and stochastic discrete measurements

A problem of constructing algorithms of optimal estimation of states of a discrete system accounting for deterministic and stochastic measurements is considered. A low order estimation algorithm is proposed in the form of a two-stage procedure for each instant of time obtained by direct application of the Kaiman optimal filtering algorithm.Translated from Dinamicheskie Sistemy, No. 4, pp. 85–89, 1985.     

Optimization of topology and shape by combining phase field modeling and discrete stochastic algorithms

We propose a concept for the comprehensive optimization of load bearing structures. The main idea is to decompose the optimization task. First, we use a phase field model to evolve the topology as basic design. The parameters of this process are the filling level of the design domain and the tendency to nucleate more or less holes. Second, we use an interface to set up a subsequent structural model, where cross-sections and positions of nodes are simultaneous optimization parameters of a metaheuristic optimization method. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Inexactly constrained discrete adjoint approach for steepest descent-based optimization algorithms

The problem of constrained optimization via the gradient-based discrete adjoint steepest descent method is studied under the assumption that the constraint equations are solved inexactly. Error propagation from the constraint equations to the gradient is studied analytically, as is the convergence rate of the inexactly constrained algorithm as it relates to the exact algorithm. A method is developed for adapting the residual tolerance to which the constraint equations are solved. The adaptive tolerance method is applied to two simple test cases to demonstrate the potential gains in computational efficiency.     

Relaxation-based algorithms for minimax optimization problems with resource allocation applications

We consider minimax optimization problems where each term in the objective function is a continuous, strictly decreasing function of a single variable and the constraints are linear. We develop relaxation-based algorithms to solve such problems. At each iteration, a relaxed minimax problem is solved, providing either an optimal solution or a better lower bound. We develop a general methodology for such relaxation schemes for the minimax optimization problem. The feasibility tests and formulation of subsequent relaxed problems can be done by using Phase I of the Simplex method and the Farkas multipliers provided by the final Simplex tableau when the corresponding problem is infeasible. Such relaxation-based algorithms are particularly attractive when the minimax optimization problem exhibits additional structure. We explore special structures for which the relaxed problem is formulated as a minimax problem with knapsack type constraints; efficient algorithms exist to solve such problems. The relaxation schemes are also adapted to solve certain resource allocation problems with substitutable resources. There, instead of Phase I of the Simplex method, a max-flow algorithm is used to test feasibility and formulate new relaxed problems.Corresponding author.Work was partially done while visiting AT&T Bell Laboratories.     

Optimal parametric synthesis of continuous/discrete stochastic terminal control systems

We propose an efficient numerical algorithm for computer parametric synthesis of linear continuous/discrete stochastic dynamic terminal control systems. The algorithm is based on the application of the method of inverse-conjugate systems and conjugate variables. Translated fromDinamicheskie Sistemy, Vol. 11, 1992.     

Methods to compare expensive stochastic optimization algorithms with random restarts

We consider the challenge of numerically comparing optimization algorithms that employ random-restarts under the assumption that only limited test data is available. We develop a bootstrapping technique to estimate the incumbent solution of the optimization problem over time as a stochastic process. The asymptotic properties of the estimator are examined and the approach is validated by an out-of-sample test. Finally, three methods for comparing the performance of different algorithms based on the estimator are proposed and demonstrated with data from a real-world optimization problem.     

Three new stochastic local search algorithms for continuous optimization problems

This paper introduces three new stochastic local search metaheuristics algorithms namely, the Best Performance Algorithm (BPA), the Iterative Best Performance Algorithm (IBPA) and the Largest Absolute Difference Algorithm (LADA). BPA and IBPA are based on the competitive nature of professional athletes, in them desiring to improve on their best recorded performances. LADA is modeled on calculating the absolute difference between two numbers. The performances of the algorithms have been tested on a large collection of benchmark unconstrained continuous optimization functions. They were benchmarked against two well-known local-search metaheuristics namely, Tabu Search (TS) and Simulated Annealing (SA). Results obtained show that each of the new algorithms delivers higher percentages of the best and mean function values found, compared to both TS and SA. The execution times of these new algorithms are also comparable. LADA gives the best performance in terms of execution time.     

Optimal subgradient algorithms for large-scale convex optimization in simple domains

This paper describes two optimal subgradient algorithms for solving structured large-scale convex constrained optimization. More specifically, the first algorithm is optimal for smooth problems with Lipschitz continuous gradients and for Lipschitz continuous nonsmooth problems, and the second algorithm is optimal for Lipschitz continuous nonsmooth problems. In addition, we consider two classes of problems: (i) a convex objective with a simple closed convex domain, where the orthogonal projection onto this feasible domain is efficiently available; and (ii) a convex objective with a simple convex functional constraint. If we equip our algorithms with an appropriate prox-function, then the associated subproblem can be solved either in a closed form or by a simple iterative scheme, which is especially important for large-scale problems. We report numerical results for some applications to show the efficiency of the proposed schemes.     

Genetic-algorithm-based simulation optimization considering a single stochastic constraint

In this paper, we consider the discrete optimization via simulation problem with a single stochastic constraint. We present two genetic-algorithm-based algorithms that adopt different sampling rules and searching mechanisms, and thus deliver different statistical guarantees. The first algorithm offers global convergence as the simulation effort goes to infinity. However, the algorithm’s finite-time efficiency may be sacrificed to maintain this theoretically appealing property. We therefore propose the second heuristic algorithm that can take advantage of the desirable mechanics of genetic algorithm, and might be better able to find near-optimal solutions in a reasonable amount of time. Empirical studies are performed to compare the efficiency of the proposed algorithms with other existing ones.