Risk and Extended Expected Utility Functions: Optimization Approaches

The proper analysis of policies under uncertainties has to deal with "hit-or-miss" type situations by using approximate risk functions, which can also be viewed as so-called extended expected utility functions. Formally this often requires the solution of dynamic stochastic optimization problems with discontinuous indicator functions of such events as ruin, underestimating costs and overestimating benefits. The available optimization techniques, in particular formulas for derivatives of risk functions, may not be applicable due to explicitly unknown probability distributions and essential discontinuities. The aim of this paper is to develop a solution technique by smoothing the risk function over certain parameters, rather than over decision variables as in the classical distribution (generalized functions) theory. For smooth approximations we obtain gradients in the form of expectations of stochastic vectors which can be viewed as a form of stochastic gradients for the original risk function. We pay special attention to optimization of risk functions defined on trajectories of discrete time stochastic processes with stopping times, which is critically important for analyzing regional vulnerability against catastrophes

The Minimization of Piecewise Functions: Pseudo Stationarity

There are many significant applied contexts that require the solution of discontinuous optimization problems in finite dimensions. Yet these problems are very difficult, both computationally and analytically. With the functions being discontinuous and a minimizer (local or global) of the problems, even if it exists, being impossible to verifiably compute, a foremost question is what kind of ''stationary solutions'' one can expect to obtain; these solutions provide promising candidates for minimizers; i.e., their defining conditions are necessary for optimality. Motivated by recent results on sparse optimization, we introduce in this paper such a kind of solution, termed ''pseudo B- (for Bouligand) stationary solution'', for a broad class of discontinuous piecewise continuous optimization problems with objective and constraint defined by indicator functions of the positive real axis composite with functions that are possibly nonsmooth. We present two approaches for computing such a solution. One approach is based on lifting the problem to a higher dimension via the epigraphical formulation of the indicator functions; this requires the addition of some auxiliary variables. The other approach is based on certain continuous (albeit not necessarily differentiable) piecewise approximations of the indicator functions and the convergence to a pseudo B-stationary solution of the original problem is established. The conditions for convergence are discussed and illustrated by an example

Mini-Workshop: Numerical Analysis for Non-Smooth PDE-Constrained Optimal Control Problems

This mini-workshop brought together leading experts working on various aspects of numerical analysis for optimal control problems with nonsmoothness. Fifteen extended abstracts summarize the presentations at this mini-workshop

Gradient-Based Estimation of Uncertain Parameters for Elliptic Partial Differential Equations

This paper addresses the estimation of uncertain distributed diffusion coefficients in elliptic systems based on noisy measurements of the model output. We formulate the parameter identification problem as an infinite dimensional constrained optimization problem for which we establish existence of minimizers as well as first order necessary conditions. A spectral approximation of the uncertain observations allows us to estimate the infinite dimensional problem by a smooth, albeit high dimensional, deterministic optimization problem, the so-called finite noise problem in the space of functions with bounded mixed derivatives. We prove convergence of finite noise minimizers to the appropriate infinite dimensional ones, and devise a stochastic augmented Lagrangian method for locating these numerically. Lastly, we illustrate our method with three numerical examples

BV solutions and viscosity approximations of rate-independent systems

In the nonconvex case solutions of rate-independent systems may develop jumps as a function of time. To model such jumps, we adopt the philosophy that rate independence should be considered as limit of systems with smaller and smaller viscosity. For the finite-dimensional case we study the vanishing-viscosity limit of doubly nonlinear equations given in terms of a differentiable energy functional and a dissipation potential which is a viscous regularization of a given rate-independent dissipation potential. The resulting definition of 'BV solutions' involves, in a nontrivial way, both the rate-independent and the viscous dissipation potential, which play a crucial role in the description of the associated jump trajectories. We shall prove a general convergence result for the time-continuous and for the time-discretized viscous approximations and establish various properties of the limiting BV solutions. In particular, we shall provide a careful description of the jumps and compare the new notion of solutions with the related concepts of energetic and local solutions to rate-independent systems