RAMP: Fundamentals, Applications and New Developments CÉSAR REGO School of Business Administration University of Mississippi University, MS 38677 USA http://faculty.bus.olemiss.edu/crego Abstract: - Relaxation Adaptive Memory Programming (RAMP) is a new metaheuristic methodology recently introduced in [4]. The method integrates the concept of Adaptive Memory Programming (AMP), originated in Tabu Search, with mathematical relaxation procedures to produce a unified framework for the design of dual and primal-dual metaheuristics. This paper reviews the fundamentals of the method and discusses major advances that have yielding exceedingly promising results in a variety of applications. Key-Words: - RAMP, Scatter Search, Surrogate Constraints, Lagrangean Relaxation, Cross-Parametric Relaxation, Subgradient Optimization, Adaptive Memory, Metaheuristics, Combinatorial Optimization. 1 Introduction Adaptive memory programming (AMP) coupled with advances in neighborhood structures derived from dynamic and adaptive search constructions have been the source of numerous important developments in metaheuristic optimization throughout the last decade. Because AMP originated in tabu search (TS), the terms TS and AMP have often been used interchangeably. However, more recently the principles of AMP have likewise been used to enhance other approaches such as genetic algorithms and evolutionary computation methods, notably including scatter search and its generalization, the path-relinking approach, whose origins share overlaps with tabu search. On the other hand, relaxation techniques have been widely used in combinatorial optimization to provide bounds for tree search procedures (such as branch and bound) as well as to produce heuristic algorithms. These techniques are based on solving an auxiliary (or relaxed) problem derived from the original by dropping or diminishing the restrictiveness of some constraints. Quite recently a new advance has occurred with the emergence of Relaxation Adaptive Memory Programming (RAMP), a method that integrates AMP with mathematical relaxation procedures to produce a unified framework for the design of dual and primal-dual metaheuristics that take full advantage of adaptive memory programming [4]. The purpose of this paper is to introduce fundamental principles of mathematical relaxation and adaptive memory programming, as embedded in the RAMP approach that is yielding new advances in the field of metaheuristics. 2 Cross-Parametric Relaxation RAMP is designed to exploit primal-dual relationships by means of different types of relaxation techniques and advanced metaheuristic strategies. As far as the dual approach is concerned, the method also takes advantage of a special relaxation technique giving rise to a cross- parametric relaxation method (CPRM), introduced in [4]. CPRM combines Lagrangean and surrogate relaxations by using a Lagrangean based subgradient search within a surrogate constraint framework to generate good surrogate constraints. In terms of graph theory, the method can be defined as using a classical subgradient search with a Lagrangean substitution as a way to produce parametric subgradients, as defined in [2]. Throughout this manuscript we define specific problems by reference to their value functions. Following this convention, consider the general 0-1 integer linear programming problem defined by ( ) { | , , {0,1}} vP Min cx Ax b Dx ex = ≤ ≤ ∈ and assume that the constraints Ax b ≤ are the ones that make the problem difficult to solve. The cross-parametric relaxation can be written as: Proceedings of the 6th WSEAS Int. Conf. on EVOLUTIONARY COMPUTING, Lisbon, Portugal, June 16-18, 2005 (pp225-227)