In this paper, we present constrained simulated annealing (CSA), a global minimization algorithm that converges to constrained global minima with probability one, for solving nonlinear discrete nonconvex constrained minimization problems. The algorithm is based on the necessary and sufficient condition for constrained local minima in the theory of discrete Lagrange multipliers we developed earlier. The condition states that the set of discrete saddle points is the same as the set of constrained local minima when all constraint functions are non-negative.

In this paper, we present an efficient global-search strategy in an algorithm based on the theory of discrete Lagrange multipliers for solving difficult SAT instances. These difficult benchmarks generally have many traps and basins that attract local-search trajectories. In contrast to trap-escaping strategies proposed earlier (?; ?) that only focus on traps, we propose a global-search strategy that penalizes a search for visiting points close to points visited before in the trajectory, where penalties are computed based on the Hamming distances between the current and historical points in the trajectory. The new

In this paper we propose an optimal anytime version of constrained simulated annealing (CSA) for solving constrained nonlinear programming problems (NLPs). One of the goals of the algorithm is to generate feasible solutions of certain prescribed quality using an average time of the same order of magnitude as that spentby the original CSA with an optimal cooling schedule in generating a solution of similar quality. Here, an optimal cooling schedule is one that leads to the shortest average total number of probes when the original CSA with the optimal schedule is run multiple times until it finds a solution. Our second goal is to design an anytime version of CSA that generates gradually improving feasible solutions as more time is spent, eventually finding a constrained global minimum (CGM). In our study,wehaveobserved a monotonically non-decreasing function relating the success probability of obtaining a solution and the average completion time of CSA, and an exponential function relating the objective target that CSA is looking for and the average completion time. Based on these observations, we have designed CSAAT;ID , the anytime CSA with iterative deepening that schedules multiple runs of CSA using a set of increasing cooling schedules and a set of improving objective targets. We then prove the optimalityofourschedules and demonstrate experimentally the results on four continuous constrained NLPs. CSAAT;ID can be generalized to solving discrete, continuous, and mixed-integer NLPs, since CSA is applicable to solve problems in these three classes. Our approach can also be generalized to other stochastic search algorithms, suchasgenetic algorithms, and be used to determine the optimal time for each run of such algorithms.

Abstract In this paper, we study the partitioning of constraints in temporal planning problems formu-lated as mixed-integer nonlinear programming (MINLP) problems. Constraint partitioning is

In this thesis, we develop constrained simulated annealing (CSA), a global optimization algorithm that asymptotically converges to constrained global minima (CGM dn ) with probability one, for solving discrete constrained nonlinear programming problems (NLPs). The algorithm is based on the necessary and sufficient condition for constrained local minima (CLM dn ) in the theory of discrete constrained optimization using Lagrange multipliers developed in our group. The theory proves the equivalence between the set of discrete saddle points and the set of CLM dn , leading to the first-order necessary and sufficient condition for CLM dn .

Nonlinear constrained optimization problems in discrete and continuous spaces are an important class of problems studied extensively in operations research and artificial intelligence. These problems can be solved by a Lagrange-multiplier method in continuous space and by an extended discrete Lagrange-multiplier method in discrete space. When constraints are satisfied, these methods rely on gradient descents in the objective space to find high-quality solutions. On the other hand, when constraints are violated, these methods rely on gradient ascents in the Lagrange-multiplier space in order to increase the penalties on unsatisfied constraints and to force these constraints into satisfaction. The balance between gradient descents and gradient ascents depends on the relative weights between the objective function and the constraints, which indirectly control the convergence speed and solution quality of the Lagrangian method. To improve convergence speed without degrading solut...

This paper studies various strategies in constrained simulated annealing (CSA), a global optimization algorithm that achieves asymptotic convergence to constrained global minima (CGM) with probability one for solving discrete constrained nonlinear programming problems (NLPs). The algorithm is based on the necessary and sufficient condition for discrete constrained local minima (CLM) in the theory of discrete Lagrange multipliers and its extensions to continuous and mixed-integer constrained NLPs. The strategies studied include adaptive neighborhoods, distributions to control sampling, acceptance probabilities, and cooling schedules. We report much better solutions than the best-known solutions in the literature on two sets of continuous benchmarks and their discretized versions.

This paper improves constrained simulated annealing (CSA), a discrete global minimization algorithm with asymptotic convergence to discrete constrained global minima with probability one. The algorithm is based on the necessary and sufficient conditions for discrete constrained local minima in the theory of discrete Lagrange multipliers. We extend CSA to solve nonlinear continuous constrained optimization problems whose variables take continuous values. We evaluate many heuristics, such as dynamic neighborhoods, gradual resolution of nonlinear equality constraints and reannealing, in order to greatly improve the efficiency of solving continuous problems. We report much better solutions than the best-known solutions in the literature on two sets of continuous optimization benchmarks. 1. Problem Definition A general constrained minimization problem is formulated as follows: minimize x f(x) (1) subject to h(x) = 0 g(x) 0 where x = (x 1 ; : : : ; xn ) is a vector of variables, f(x) is a...

This chapter presents a framework that unifies various search mechanisms for solving constrained nonlinear programming (NLP) problems. These problems are characterized by functions that are not necessarily differentiable and continuous. Our proposed framework is based on the first-order necessary and sufficient condition developed for constrained local minimization in discrete space that shows the equivalence between discrete-neighborhood saddle points and constrained local minima. To look for discrete-neighborhood saddle points, we formulate a discrete constrained NLP in an augmented Lagrangian function and study various mechanisms for performing ascents of the augmented function in the original-variable subspace and descents in the Lagrange-multiplier subspace. Our results show that CSAGA, a combined constrained simulated annealing and genetic algorithm, performs well when using crossovers, mutations, and annealing to generate trial points. Finally, we apply iterative deepening to de...

In this paper, we propose new dominance relations that can speed up significantly the solution process of planning problems formulated as nonlinear constrained dynamic optimization in discrete time and space. We first show that path dominance in dynamic programming cannot be applied when there are general constraints that span across multiple stages, and that node dominance, in the form of Euler-Lagrange conditions developed in optimal control theory in continuous space, cannot be extended to that in discrete space. This paper is the first to propose efficient node-dominance relations, in the form of local saddle-point conditions in each stage of a discrete-space planning problem, for pruning states that will not lead to locally optimal paths. By utilizing these dominance relations, we present efficient search algorithms whose complexity, despite exponential, has a much smaller base as compared to that without using the relations. Finally, we demonstrate the performance of our approach by integrating it in the ASPEN planner and show significant improvements in CPU time and solution quality on some spacecraft scheduling and planning benchmarks.