Showing papers by "Center for Discrete Mathematics and Theoretical Computer Science published in 2013"

A penalty function-based differential evolution algorithm for constrained global optimization

Abstract: We propose a differential evolution-based algorithm for constrained global optimization. Although differential evolution has been used as the underlying global solver, central to our approach is the penalty function that we introduce. The adaptive nature of the penalty function makes the results of the algorithm mostly insensitive to low values of the penalty parameter. We have also demonstrated both empirically and theoretically that the high value of the penalty parameter is detrimental to convergence, specially for functions with multiple local minimizers. Hence, the penalty function can dispense with the penalty parameter. We have extensively tested our penalty function-based DE algorithm on a set of 24 benchmark test problems. Results obtained are compared with those of some recent algorithms.

Max-k-Cut by the Discrete Dynamic Convexized Method

Abstract: In this paper, we propose a “multistart-type” algorithm for solving the max-k-cut problem. Central to our algorithm is an auxiliary function we propose. We formulate the max-k-cut problem as an explicit mathematical form, which allows us to use an easy implementable local search. The construction of the auxiliary function requires a local maximizer of the max-k-cut problem. If the best local maximizer obtained is used in the construction of the auxiliary function, then the local maximization of the auxiliary function leads to a better maximizer of the max-k-cut problem. This proves to be a good strategy to escape from the current local optima and to search a broader solution space. Indeed, we have shown, both numerically and theoretically, that the maximization of the auxiliary function by the local search method can escape successfully from previously converged discrete local maximizers by taking increasing values of a parameter. Computational results on many test instances with different sizes and densities show that the proposed algorithm is efficient and stable to find approximate global solutions for the max-k-cut problems. Although we have presented results for k ≥ 2, the robustness of our algorithm is shown for k = 2 by comparisons with a number of recent methods. A number of theoretical results are also presented, which justify the design of our algorithm.

A Latent Source Model to Detect Multiple Spatial Clusters With Application in a Mobile Sensor Network for Surveillance of Nuclear Materials

Abstract: Potential nuclear attacks are among the most devastating terrorist attacks, with severe loss of human lives as well as damage to infrastructure. To deter such threats, it becomes increasingly vital to have sophisticated nuclear surveillance and detection systems deployed in major cities in the United States, such as New York City. In this article, we design a mobile sensor network and develop statistical algorithms and models to provide consistent and pervasive surveillance of nuclear materials in major cities. The network consists of a large number of vehicles on which nuclear sensors and Global Position System (GPS) tracking devices are installed. Real time sensor readings and GPS information are transmitted to and processed at a central surveillance center. Mathematical and statistical analyses are performed, in which we mimic a signal-generating process and develop a latent source modeling framework to detect multiple spatial clusters. A Monte Carlo expectation-maximization algorithm is developed to e...

On the Asymptotic Statistics of the Number of Occurrences of Multiple Permutation Patterns

Abstract: We study statistical properties of the random variables $X_{\sigma}(\pi)$, the number of occurrences of the pattern $\sigma$ in the permutation $\pi$. We present two contrasting approaches to this problem: traditional probability theory and the ``less traditional'' computational approach. Through the perspective of the first one, we prove that for any pair of patterns $\sigma$ and $\tau$, the random variables $X_{\sigma}$ and $X_{\tau}$ are jointly asymptotically normal (when the permutation is chosen from $S_{n}$). From the other perspective, we develop algorithms that can show asymptotic normality and joint asymptotic normality (up to a point) and derive explicit formulas for quite a few moments and mixed moments empirically, yet rigorously. The computational approach can also be extended to the case where permutations are drawn from a set of pattern avoiders to produce many empirical moments and mixed moments. This data suggests that some random variables are not asymptotically normal in this setting.

A discrete dynamic convexized method for VLSI circuit partitioning

Abstract: In this paper, we consider the circuit partitioning problem, which is a fundamental problem in computer-aided design of very large-scale-integrated circuits. We formulate the problem as an equivalent constrained integer programming problem by constructing an auxiliary function. A global search method, entitled the dynamic convexized method, is developed for the integer programming problem. We modify the Fiduccia–Mattheyses FM algorithm, which is a fundamental partitioning algorithm for the circuit partitioning problem, to minimize the auxiliary function. We show both computationally and theoretically that our method can escape successfully from previous discrete local minimizers by taking increasing values of a parameter. Experimental results on ACM/SIGDA and ISPD98 benchmarks show up to 58% improvements over the well-known FM algorithm in terms of the best cutsize. Furthermore, we integrate the algorithm with the state-of-the-art practical multilevel partitioner MLPart. Experiments on the same set of benchmarks show that the solutions obtained in this way has 3–7% improvements over that of the MLPart.