Optimization Strategies in Complex Systems

TL;DR

A class of combinatorial optimization problems that emerge in a variety of domains among which: condensed matter physics, theory of financial risks, error correcting codes in information transmissions, molecular and protein conformation, image restoration is considered.

Abstract

We consider a class of combinatorial optimization problems that emerge in a variety of domains among which: condensed matter physics, theory of financial risks, error correcting codes in information transmissions, molecular and protein conformation, image restoration. We show the performances of two algorithms, the``greedy'' (quick decrease along the gradient) and the``reluctant'' (slow decrease close to the level curves) as well as those of a``stochastic convex interpolation''of the two. Concepts like the average relaxation time and the wideness of the attraction basin are analyzed and their system size dependence illustrated.

Figures (3)

• Figure 1: The average time to reach a metastable configuration for different values of $P$. Top to bottom: $P = 0$ (reluctant), $P=0.1$, $P=0.5$, $P=0.9$, $P=1$ (greedy). The continuous lines are the numerical fits to power law: $\tau(N) \sim N^{\alpha}$, with $\alpha = 2.07, 1.26, 1.08, 1.05, 1.04$ from top to bottom.
• Figure 2: Lowest energy value for a fixed number of $N$ initial conditions for different value of $P$. Bottom to top: $P = 0$ (reluctant), $P=0.1$, $P=0.5$, $P=0.9$, $P=1$ (greedy)
• Figure 3: Lowest energy value for a fixed elapsed time of $100$ hours (each run) on a IBM SP3 for different value of $P$ (see legend)