The Optimality of Decision Making During Motor Learning

In our daily lives, we often must predict how well we are going to perform in the future based on an evaluation of our current performance and an assessment of how much we will improve with practice. Such predictions can be used to decide whether to invest our time and energy in learning and, if we opt to invest, what rewards we may gain. This thesis investigated whether people are capable of tracking their own learning (i.e. current and future motor ability) and exploiting that information to make decisions related to task reward. In experiment one, participants performed a target aiming task under a visuomotor rotation such that they initially missed the target but gradually improved. After briefly practicing the task, they were asked to select rewards for hits and misses applied to subsequent performance in the task, where selecting a higher reward for hits came at a cost of receiving a lower reward for misses. We found that participants made decisions that were in the direction of optimal and therefore demonstrated knowledge of future task performance. In experiment two, participants learned a novel target aiming task in which they were rewarded for target hits. Every five trials, they could choose a target size which varied inversely with reward value. Although participants’ decisions deviated from optimal, a model suggested that they took into account both past performance, and predicted future performance, when making their decisions. Together, these experiments suggest that people are capable of tracking their own learning and using that information to make sensible decisions related to reward maximization.

New Rigorous Decomposition Methods for Mixed-integer Linear and Nonlinear Programming

Process systems design, operation and synthesis problems under uncertainty can readily be formulated as two-stage stochastic mixed-integer linear and nonlinear (nonconvex) programming (MILP and MINLP) problems. These problems, with a scenario based formulation, lead to large-scale MILPs/MINLPs that are well structured. The first part of the thesis proposes a new finitely convergent cross decomposition method (CD), where Benders decomposition (BD) and Dantzig-Wolfe decomposition (DWD) are combined in a unified framework to improve the solution of scenario based two-stage stochastic MILPs. This method alternates between DWD iterations and BD iterations, where DWD restricted master problems and BD primal problems yield a sequence of upper bounds, and BD relaxed master problems yield a sequence of lower bounds. A variant of CD, which includes multiple columns per iteration of DW restricted master problem and multiple cuts per iteration of BD relaxed master problem, called multicolumn-multicut CD is then developed to improve solution time. Finally, an extended cross decomposition method (ECD) for solving two-stage stochastic programs with risk constraints is proposed. In this approach, a CD approach at the first level and DWD at a second level is used to solve the original problem to optimality. ECD has a computational advantage over a bilevel decomposition strategy or solving the monolith problem using an MILP solver. The second part of the thesis develops a joint decomposition approach combining Lagrangian decomposition (LD) and generalized Benders decomposition (GBD), to efficiently solve stochastic mixed-integer nonlinear nonconvex programming problems to global optimality, without the need for explicit branch and bound search. In this approach, LD subproblems and GBD subproblems are systematically solved in a single framework. The relaxed master problem obtained from the reformulation of the original problem, is solved only when necessary. A convexification of the relaxed master problem and a domain reduction procedure are integrated into the decomposition framework to improve solution efficiency. Using case studies taken from renewable resource and fossil-fuel based application in process systems engineering, it can be seen that these novel decomposition approaches have significant benefit over classical decomposition methods and state-of-the-art MILP/MINLP global optimization solvers.