Summary: A variety of strategies have been proposed for overcoming local optimality in metaheuristic search. This paper examines characteristics of moves that can be exploited to make good decisions about steps that lead away from a local optimum and then lead toward a new local optimum. We introduce strategies to identify and take advantage of useful features of solution history with an adaptive memory metaheuristic, to provide rules for selecting moves that offer promise for discovering improved local optima. Our approach uses a new type of adaptive memory based on a construction called exponential extrapolation. The memory operates by means of threshold inequalities that ensure selected moves will not lead to a specified number of most recently encountered local optima. Associated thresholds are embodied in choice rule strategies that further exploit the exponential extrapolation concept and open a variety of research possibilities for exploration. The considerations treated in this study are illustrated in an implementation to solve the Quadratic Unconstrained Binary Optimization (QUBO) problem. We show that the AA algorithm obtains an average objective gap of 0.0315% to the best known values for the 21 largest Palubeckis instances. This solution quality is considered to be quite attractive because less than 20 s on average are taken by AA, which is 1 to 2 orders of magnitude less than the time required by most algorithms reporting the best known results.