Jasper C. H. Lee — Page 2

arXiv2018

Augmenting Stream Constraint Programming with Eventuality Conditions

Stream constraint programming is a recent addition to the family of constraint programming frameworks, where variable domains are sets of infinite streams over finite alphabets. Previous works showed promising results for its applicability to real-world planning and control problems. In this paper, motivated by the modelling of planning applications, we improve the expressiveness of the framework by introducing 1) the "until" constraint, a new construct that is adapted from Linear Temporal Logic and 2) the @ operator on streams, a syntactic sugar for which we provide a more efficient solving algorithm over simple desugaring. For both constructs, we propose corresponding novel solving algorithms and prove their correctness. We present competitive experimental results on the Missionaries and Cannibals logic puzzle and a standard path planning application on the grid, by comparing with Apt and Brand's method for verifying eventuality conditions using a CP approach.

cs.AI↗

arXiv2016

Optimizing Star-Convex Functions

We introduce a polynomial time algorithm for optimizing the class of star-convex functions, under no restrictions except boundedness on a region about the origin, and Lebesgue measurability. The algorithm's performance is polynomial in the requested number of digits of accuracy, contrasting with the previous best known algorithm of Nesterov and Polyak that has exponential dependence, and that further requires Lipschitz second differentiability of the function, but has milder dependence on the dimension of the domain. Star-convex functions constitute a rich class of functions generalizing convex functions to new parameter regimes, and which confound standard variants of gradient descent; more generally, we construct a family of star-convex functions where gradient-based algorithms provably give no information about the location of the global optimum. We introduce a new randomized algorithm for finding cutting planes based only on function evaluations, where, counterintuitively, the algorithm must look outside the feasible region to discover the structure of the star-convex function that lets it compute the next cut of the feasible region. We emphasize that the class of star-convex functions we consider is as unrestricted as possible: the class of Lebesgue measurable star-convex functions has theoretical appeal, introducing to the domain of polynomial-time algorithms a huge class with many interesting pathologies. We view our results as a step forward in understanding the scope of optimization techniques beyond the garden of convex optimization and local gradient-based methods.

cs.DS↗