A framework for analyzing sub-optimal performance of local search algorithms

This paper presents a framework for analyzing and comparing sub-optimal performance of local search algorithms for hard discrete optimization problems. The β-acceptable solution probability is introduced that captures how effectively an algorithm has performed to date and how effectively an algorithm can be expected to perform in the future. Using this probability, the necessary conditions for a local search algorithm to converge in probability to β-acceptable solutions are derived. To evaluate and compare the effectiveness of local search algorithms, two estimators for the expected number of iterations to visit a β-acceptable solution are obtained. Computational experiments are reported with simulated annealing and tabu search applied to four small traveling salesman problem instances, and the Lin-Kernighan-Helsgaun algorithm applied to eight medium to large traveling salesman problem instances (all with known optimal solutions), to illustrate the application of these estimators.     

Approximation algorithms for MAX RES CUT with limited unbalanced constraints

Two kinds of MAX RES CUT problems, the MAX s?t CUT and the MAX s?t?v CUT, with limited unbalanced constraints are considered. Approximation algorithms used in Frieze and Jerrum (Integer Programming and Combinatorial Optimization, vol. 920, pp. 1–13, Springer, Berlin, 1995), Galbiati and Maffioli (Theor. Comput. Sci. 385:78–87, 2007), Han et al. (Math. Program. Ser. B 92:509–535, 2002) and Ye (Math. Programm. 90:101–111, 2001) are extended to the two MAX RES CUT problems. A special matrix P is constructed by which it can ensure that the given nodes s,t are feasible to equality constraints with probability one for the MAX s?t CUT and s,t,v are feasible to equality constraints with at least probability 0.912 for the MAX s?t?v CUT. A fussy greedy sizing-adjusted procedure is then proposed to confirm that the round solution is feasible for all constraints. We find these extensions are nontrivial and some interesting results about performance ratio are obtained for the MAX RES CUT problem with limited unbalanced constraints.     

A novel chaotic particle swarm optimization approach using Hénon map and implicit filtering local search for economic load dispatch

Particle swarm optimization (PSO) is a population-based swarm intelligence algorithm driven by the simulation of a social psychological metaphor instead of the survival of the fittest individual. Based on the chaotic systems theory, this paper proposed a novel chaotic PSO combined with an implicit filtering (IF) local search method to solve economic dispatch problems. Since chaotic mapping enjoys certainty, ergodicity and the stochastic property, the proposed PSO introduces chaos mapping using Hénon map sequences which increases its convergence rate and resulting precision. The chaotic PSO approach is used to produce good potential solutions, and the IF is used to fine-tune of final solution of PSO. The hybrid methodology is validated for a test system consisting of 13 thermal units whose incremental fuel cost function takes into account the valve-point loading effects. Simulation results are promising and show the effectiveness of the proposed approach.     

Speeding up IP-based algorithms for constrained quadratic 0–1 optimization

In many practical applications, the task is to optimize a non-linear objective function over the vertices of a well-studied polytope as, e.g., the matching polytope or the travelling salesman polytope (TSP). Prominent examples are the quadratic assignment problem and the quadratic knapsack problem; further applications occur in various areas such as production planning or automatic graph drawing. In order to apply branch-and-cut methods for the exact solution of such problems, the objective function has to be linearized. However, the standard linearization usually leads to very weak relaxations. On the other hand, problem-specific polyhedral studies are often time-consuming. Our goal is the design of general separation routines that can replace detailed polyhedral studies of the resulting polytope and that can be used as a black box. As unconstrained binary quadratic optimization is equivalent to the maximum-cut problem, knowledge about cut polytopes can be used in our setting. Other separation routines are inspired by the local cuts that have been developed by Applegate, Bixby, Chvátal and Cook for faster solution of large-scale traveling salesman instances. Finally, we apply quadratic reformulations of the linear constraints as proposed by Helmberg, Rendl and Weismantel for the quadratic knapsack problem. By extensive experiments, we show that a suitable combination of these methods leads to a drastic speedup in the solution of constrained quadratic 0–1 problems. We also discuss possible generalizations of these methods to arbitrary non-linear objective functions.     

Subgradient projection algorithms for constrained nonsmooth optimization II: nonlinear constraints†

In this paper, a kind of subgradient projection algorithms is established for minimizing a locally Lipschitz continuous function subject to nonlinearly smooth constraints, which is based on the idea to get a feasible and strictly descent direction by combining the ?-subgradient projection direction that attempts to satisfy the Kuhn-Tucker conditions with one corrected direction produced by a linear programming subproblem. The algorithm avoids the zigzagging phenomenon and converges to Kuhn-Tucker points, due to using the c.d.f. maps of Polak and Mayne (1985), ?active constraints and ?adjusted rules     

Are there elimination algorithms for the permanent?

We define the class of elimination algorithms. There are algebraic algorithms for evaluating multivariate polynomials, and include as a special case Gaussian elimination for evaluating the determinant. We show how to find the linear symmetries of a polynomial, defined appropriately, and use these methods to find the linear symmetries of the permanent and determinant. We show that in contrast to the Gaussian elimination algorithm for the determinant, there is no elimination algorithm for the permanent.     

Hardness measures for gridworld benchmarks and performance analysis of real-time heuristic search algorithms

Gridworlds are one of the most popular settings used in benchmark problems for real-time heuristic search algorithms. However, no comprehensive studies have been published so far on how the difference in the density of randomly positioned obstacles affects the hardness of the problems. This paper presents two measures for characterizing the hardness of gridworld problems parameterized by obstacle ratio, and relates them to the performance of the algorithms. We empirically show that the peak locations of those measures and actual performance degradation of the basic algorithms (RTA* and LRTA*) almost coincide with each other for a wide variety of problem settings. Thus the measures uncover some interesting aspects of the gridworlds.     

Effective formulas for the local ?ojasiewicz exponent

We give an effective formula for the local ?ojasiewicz exponent of a polynomial mapping. Moreover, we give an algorithm for computing the local dimension of an algebraic variety.     

Spectral instability for some Schr?dinger operators

Summary. We show that the condition numbers of isolated eigenvalues of typical non-self-adjoint differential operators such as the harmonic oscillator may be extremely large. We describe a stable procedure for computing the condition numbers for Schr?dinger operators in one dimension, and apply it to the complex resonances of a typical operator with a dilation analytic potential. Received October 9, 1998 / Revised version received September 13, 1999 / Published online 16 March 2000     

Hybridized evolutionary local search algorithm for the team orienteering problem with time windows

The orienteering problem (OP) consists in finding an elementary path over a subset of vertices. Each vertex is associated with a profit that is collected on the visitor’s first visit. The objective is to maximize the collected profit with respect to a limit on the path’s length. The team orienteering problem (TOP) is an extension of the OP where a fixed number m of paths must be determined. This paper presents an effective hybrid metaheuristic to solve both the OP and the TOP with time windows. The method combines the greedy randomized adaptive search procedure (GRASP) with the evolutionary local search (ELS). ELS generates multiple distinct child solutions using a mutation mechanism. Each child solution is further improved by a local search procedure. GRASP provides multiple starting solutions to the ELS. The method is able to improve several best known results on available benchmark instances.     

Continuous ant colony system and tabu search algorithms hybridized for global minimization of continuous multi-minima functions

A new hybrid optimization method, combining Continuous Ant Colony System (CACS) and Tabu Search (TS) is proposed for minimization of continuous multi-minima functions. The new algorithm incorporates the concepts of promising list, tabu list and tabu balls from TS into the framework of CACS. This enables the resultant algorithm to avoid bad regions and to be guided toward the areas more likely to contain the global minimum. New strategies are proposed to dynamically tune the radius of the tabu balls during the execution and also to handle the variable correlations. The promising list is also used to update the pheromone distribution over the search space. The parameters of the new method are tuned based on the results obtained for a set of standard test functions. The results of the proposed scheme are also compared with those of some recent ant based and non-ant based meta-heuristics, showing improvements in terms of accuracy and efficiency.     

Iterated local search with Powell’s method: a memetic algorithm for continuous global optimization

In combinatorial solution spaces Iterated Local Search (ILS) turns out to be exceptionally successful. The question arises: is ILS also capable of improving the optimization process in continuous solution spaces? To demonstrate that hybridization leads to powerful techniques in continuous domains, we introduce a hybrid meta-heuristic that integrates Powell’s direct search method. It combines direct search with elements from population based evolutionary optimization. The approach is analyzed experimentally on a set of well known test problems and compared to a state-of-the-art technique, i.e., a restart variant of the Covariance Matrix Adaptation Evolution Strategy with increasing population sizes (G-CMA-ES). It turns out that the population-based Powell-ILS is competitive to the CMA-ES, in some cases even significantly faster and behaves more robust than the pure strategy of Powell in multimodal fitness landscapes. Further experiments on the perturbation mechanism, population sizes, and problems with noise complete the analysis of the hybrid methodology and lead to parameter recommendations.     

Metaheuristic methods based on Tabu search for assigning judges to competitions

Two metaheuristic methods based on Tabu search are introduced to assign judges to individual competitions in a tournament. The complexity of the mathematical formulation accounting for the assignment rules, leads us to use such an approach. The first metaheuristic includes two different Tabu searches that are combined with a diversification strategy. The second metaheuristic is applied to a penalized version of the original model formulated as an assignment problem. This metaheuristic is also based on a Tabu search procedure including a diversification strategy driven by the constraints violated. Numerical results are provided to indicate the efficiency of the methods to generate very good solutions.     

Stabilization to a positive equilibrium for some reaction–diffusion systems

The aim of this paper is to study the asymptotic behavior of solutions for some reaction–diffusion systems in biology. First, we establish a Liouville type theorem for entire solutions of these reaction–diffusion systems. Based on this theorem, we derive the stabilization of the solutions of the reaction–diffusion system to the unique positive constant state, under the condition that this positive constant state is globally stable in the corresponding kinetic systems. Two specific examples about spreading phenomena from ecology and epidemiology are given to illustrate the application of this theory.     

The Poincaré series of a local Gorenstein ring of multiplicity up to 10 is rational

Let R be a local, Gorenstein ring with algebraically closed residue field k of characteristic 0 and let P _R (z):= Σ _p=0^∞ dim _k (Tor _p ^R (k, k))z ^p be its Poincaré series. We compute P _R when R belongs to a particular class defined in the Introduction, proving its rationality. As a by-product we prove the rationality of P _R for all local, Gorenstein rings of multiplicity at most 10.     

A distribution of the number of ℓ-steps in a random binary sequence subject to some constraints

There are investigated the joint distribution of random variables _kn(₁),..., _kn(_s), and distributions of some functionals of _kn(), for n. Here _kn(), 1ln–1 is the number of -steps in a binary sequence (b.s.), selected randomly and equiprobably from the totality of all n-dimensional b.s. that have a prescribed number of ones and k 1-steps. By an -step of a b.s. we understand a configuration of the form 1...0, where the ellipsis stands for an ( –1)-dimensional b.s.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 9, pp. 1186–1193, September, 1991.     

Discrete and geometric Branch and Bound algorithms for?medical image registration

Aiming at the development of an exact solution method for registration problems, we present two different Branch & Bound algorithms for a mixed integer programming formulation of the problem. The first B&B algorithm branches on binary assignment variables and makes use of an optimality condition that is derived from a graph matching formulation. The second, geometric B&B algorithm applies a geometric branching strategy on continuous transformation variables. The two approaches are compared for synthetic test examples as well as for 2-dimensional medical data. The results show that medium sized problem instances can be solved to global optimality in a reasonable amount of time.     

From pointwise to local regularity for solutions of Hamilton–Jacobi equations

It is well-known that solutions to the Hamilton–Jacobi equation $$\begin{aligned} u_{t}(t,x)+H(x,u_{x}(t,x))=0 \end{aligned}$$ fail to be everywhere differentiable. Nevertheless, suppose a solution $u$ turns out to be differentiable at a given point $(t,x)$ in the interior of its domain. May then one deduce that $u$ must be continuously differentiable in a neighborhood of $(t,x)$ ? Although this question has a negative answer in general, our main result shows that it is indeed the case when the proximal subdifferential of $u(t,\cdot )$ at $x$ is nonempty. Our approach uses the representation of $u$ as the value function of a Bolza problem in the calculus of variations, as well as necessary conditions for such a problem.