For a broad class of nonlinear systems, we formulate the problem guaranteeing safety with optimality under constraints. Specifically, define controlled for differential inclusions constraints on states and inputs. Through use nonsmooth analysis tools, show that continuous optimal control law can be selected from set-valued constraint capturing system conditions using barrier functions. Our resu...

We present a new algorithm, called manifold sampling, for the unconstrained minimization of a nonsmooth composite function h ◦ F when h has known structure. In particular, by classifying points in the domain of the nonsmooth function h into manifolds, we adapt search directions within a trust-region framework based on knowledge of manifolds intersecting the current trust region. We motivate thi...

We present a new algorithm, called manifold sampling, for the unconstrained minimization of a nonsmooth composite function h ◦ F . By classifying points in the domain of the nonsmooth function h into what we call manifolds, we adapt search directions within a trust-region framework based on knowledge of manifolds intersecting the current trust region. We motivate this idea through a study of l1...

The problem of minimizing the sum of nonsmooth, convex objective functions defined on a real Hilbert space over the intersection of fixed point sets of nonexpansive mappings, onto which the projections cannot be efficiently computed, is considered. The use of proximal point algorithms that use the proximity operators of the objective functions and incremental optimization techniques is proposed...

Kavanagh, Kathleen Rose. Nonsmooth Nonlinearities in Applications From Hydrology. (Under the direction of C.T. Kelley) This work has two parts; simulation of unsaturated flow and optimization of remediation problems. For the unsaturated flow simulation, we propose an adaptive time stepping scheme based on error control for Richards’ equation, a model for flow in unsaturated porous media. The mo...

We present a new class of decentralized first-order methods for nonsmooth and stochastic optimization problems defined over multiagent networks. Considering that communication is a major bottleneck in decentralized optimization, our main goal in this paper is to develop algorithmic frameworks which can significantly reduce the number of inter-node communications. We first propose a decentralize...

In this article, we present inverse and implicit function theorems forH -differentiable functions and semismooth functions, thereby generalizing the classical inverse and implicit function theorems to certain classes of nonsmooth functions. The classical inverse function theorem [47] asserts that a continuously differentiable function f : R → R is locally invertible at a point with a continuous...

The paper is devoted to the subdifferential study and applications of the supremum of uniformly Lipschitzian functions over arbitrary index sets with no topology. Based on advanced techniques of variational analysis, we evaluate major subdifferentials of the supremum functions in the general framework of Asplund (in particular, reflexive) spaces with no convexity or relaxation assumptions. The ...