The optimality conditions of a nonlinear second-order cone program can be reformulated as a nonsmooth system of equations using a projection mapping. This allows the application of nonsmooth Newton methods for the solution of the nonlinear second-order cone program. Conditions for the local quadratic convergence of these nonsmooth Newton methods are investigated. Related conditions are also giv...

In this paper we develop a novel criterion for choosing regularization parameters for nonsmooth Tikhonov functionals. The proposed criterion is solely based on the value function, and thus applicable to a broad range of functionals. It is analytically compared with the local minimum criterion, and a posteriori error estimates are derived. An efficient numerical algorithm for computing the minim...

In this thesis, we focus on the Nonsmooth Boundary-valued ODEs, whose right-hand functions are parameterized by algebraic variables that solve the initial-valuedproblems and the nonsmooth equations. Based on the idea of the initial value tech-niques used in traditional ODEs, a smoothing Newton method originated from theQSZ method is applied to the nonmsmooth dynamic systems. Mos...

In this paper we introduce a new primal-dual technique for convergence analysis of gradient schemes for nonsmooth convex optimization. As an example of its application, we derive a primal-dual gradient method for a special class of structured nonsmooth optimization problems, which ensures a rate of convergence of order O( 1 k ), where k is the iteration count. Another example is a gradient sche...

In this paper a new algorithm to locally minimize nonsmooth, nonconvex functions is developed. We introduce the notion of secants and quasisecants for nonsmooth functions. The quasisecants are applied to find descent directions of locally Lipschitz functions. We design a minimization algorithm which uses quasisecants to find descent directions. We prove that this algorithm converges to Clarke s...

This paper is devoted to the theoretical and numerical study of an optimal design problem in high-temperature superconductivity (HTS). The shape optimization find superconductor minimizes a certain cost functional under given target on electric field over specific domain interest. For governing PDE-model, we consider elliptic curl-curl variational inequality (VI) second kind with L1-type nonlin...

This study debuts a new spline dimensional decomposition (SDD) for uncertainty quantification analysis of high-dimensional functions, including those endowed with high nonlinearity and nonsmoothness, if they exist, in proficient manner. The creates hierarchical expansion an output random variable interest respect to measure-consistent orthonormalized basis splines (B-splines) independent input ...

Passivity and dissipativity, in Jan Willems’s sense, are known to be powerful tools for stability analysis feedback control design. For instance, passive systems negative interconnection with slope-restricted, static, single-valued, smooth nonlinearities, as Lur’e systems, have been thoroughly studied automatic control, yielding the so-called absolute problem (with famous Popov circle criteria)...

In this paper, we consider a simple bilevel program where the lower level program is a nonconvex minimization problem with a convex set constraint and the upper level program has a convex set constraint. By using the value function of the lower level program, we reformulate the bilevel program as a single level optimization problem with a nonsmooth inequality constraint and a convex set constra...

We consider in this paper a class of composite optimization problems whose objective function is given by the summation of a general smooth and nonsmooth component, together with a relatively simple nonsmooth term. We present a new class of first-order methods, namely the gradient sliding algorithms, which can skip the computation of the gradient for the smooth component from time to time. As a...