Fundamental Principles of Control Landscapes with Applications to Quantum Mechanics, Chemistry and Evolution

Abstract

Abstract. The concept of a landscape or response surface naturally arises in applications widely ranging over the sciences, engineering and other disciplines. A landscape is the desired output as a function of a set of input variables, often of very high dimension. The relationship between the features of a landscape and the input variables is usually unknown a priori and often thought to be highly complex due to the anticipated intricate interactions involved. This chapter reviews recent developments in the analysis of landscape topology with the input variables considered as controls. Taking a control perspective allows for the specification of particular assumptions whose satisfaction permits a general analysis of the landscape topology. Satisfaction of these conditions leads to the conclusion that control landscapes should be devoid of suboptimal critical point traps, thereby permitting ready excursions without hindrance to the highest values of the landscape. These principles are set out in a general framework and then specifically illustrated for applications

The Surprising Ease of Finding Optimal Solutions for Controlling Nonlinear Phenomena in Quantum and Classical Complex Systems

“…Nonlinear phenomena cover vast areas in the sciences and beyond, and it is an open question about which applications will satisfy the same general N-CL principle. A primary question concerning N-CLs 9,11 is whether local optima exist forming traps, as shown in Figure 1, compared to the trap-free situation in Figure 2. These two figures are simply schematics with two control variables, u = [u 1 , u 2 ], along the horizontal axes and with the objective fitness value indicated by the height domains (e.g., directed and natural evolutions) do not share this feature, at least at small scales over the associated landscapes where some degree of roughness can be expected.…”

Optimization Landscape of Quantum Control Systems

“…2 for illustration), which collectively show that the control landscape is trap free under appropriate conditions. These works have been reviewed in several survey papers [7,8,14,15] that also reveal the rich structures of saddle manifolds in many of the landscapes. This review will survey the landscape studies in quantum control that cover both fundamental results and most recent advances.…”

Scale-invariance of ruggedness measures in fractal fitness landscapes

“…For some application domains the decision variables may be payoff attached to evolutionary games, as in evolutionary game theory defining payoff landscapes [5], or trading strategies as in financial market analysis, which entails profit landscapes [11]. Further examples of decision variables are conformations of molecular entities or spatial positions of interacting molecules as in the theory of spin glasses or folding and energy relaxation in proteins and nucleic acids, which uses energy landscapes [2,24,40], or cluster structures in large data analysis, which employs cost landscapes [10], or control variables of electromagnetic fields as in quantum dynamics, which defines control landscapes [27,28]. Lastly, and arguable most prominently, the variables may be genotypes or search space elements, as in evolutionary biology and evolutionary computation defining fitness landscapes [9,36,34].…”