Non deterministic polynomial optimization problems and their approximations

AbstractA unified and general framework for the study of nondeterministic polynomial optimization problems (NPOP) is presented and some properties of NPOP's are investigated. A characterization of NPOP's with regard to their approximability properties is given by proving necessary and sufficient conditions for two approximability schemes. Known approximability results are shown to fit within the general frame developed in the paper. Finally NPOP's are classified and studied with regard to the possibility or impossibility of ‘reducing’ certain types of NPOP's to other types in a sense specified in the text

High-Dimensional Stochastic Design Optimization by Adaptive-Sparse Polynomial Dimensional Decomposition

This paper presents a novel adaptive-sparse polynomial dimensional decomposition (PDD) method for stochastic design optimization of complex systems. The method entails an adaptive-sparse PDD approximation of a high-dimensional stochastic response for statistical moment and reliability analyses; a novel integration of the adaptive-sparse PDD approximation and score functions for estimating the first-order design sensitivities of the statistical moments and failure probability; and standard gradient-based optimization algorithms. New analytical formulae are presented for the design sensitivities that are simultaneously determined along with the moments or the failure probability. Numerical results stemming from mathematical functions indicate that the new method provides more computationally efficient design solutions than the existing methods. Finally, stochastic shape optimization of a jet engine bracket with 79 variables was performed, demonstrating the power of the new method to tackle practical engineering problems.Comment: 18 pages, 2 figures, to appear in Sparse Grids and Applications--Stuttgart 2014, Lecture Notes in Computational Science and Engineering 109, edited by J. Garcke and D. Pfl\"{u}ger, Springer International Publishing, 201

Pricing American Options by Exercise Rate Optimization

We present a novel method for the numerical pricing of American options based on Monte Carlo simulation and the optimization of exercise strategies. Previous solutions to this problem either explicitly or implicitly determine so-called optimal exercise regions, which consist of points in time and space at which a given option is exercised. In contrast, our method determines the exercise rates of randomized exercise strategies. We show that the supremum of the corresponding stochastic optimization problem provides the correct option price. By integrating analytically over the random exercise decision, we obtain an objective function that is differentiable with respect to perturbations of the exercise rate even for finitely many sample paths. The global optimum of this function can be approached gradually when starting from a constant exercise rate. Numerical experiments on vanilla put options in the multivariate Black-Scholes model and a preliminary theoretical analysis underline the efficiency of our method, both with respect to the number of time-discretization steps and the required number of degrees of freedom in the parametrization of the exercise rates. Finally, we demonstrate the flexibility of our method through numerical experiments on max call options in the classical Black-Scholes model, and vanilla put options in both the Heston model and the non-Markovian rough Bergomi model

Linearly Solvable Stochastic Control Lyapunov Functions

This paper presents a new method for synthesizing stochastic control Lyapunov functions for a class of nonlinear stochastic control systems. The technique relies on a transformation of the classical nonlinear Hamilton-Jacobi-Bellman partial differential equation to a linear partial differential equation for a class of problems with a particular constraint on the stochastic forcing. This linear partial differential equation can then be relaxed to a linear differential inclusion, allowing for relaxed solutions to be generated using sum of squares programming. The resulting relaxed solutions are in fact viscosity super/subsolutions, and by the maximum principle are pointwise upper and lower bounds to the underlying value function, even for coarse polynomial approximations. Furthermore, the pointwise upper bound is shown to be a stochastic control Lyapunov function, yielding a method for generating nonlinear controllers with pointwise bounded distance from the optimal cost when using the optimal controller. These approximate solutions may be computed with non-increasing error via a hierarchy of semidefinite optimization problems. Finally, this paper develops a-priori bounds on trajectory suboptimality when using these approximate value functions, as well as demonstrates that these methods, and bounds, can be applied to a more general class of nonlinear systems not obeying the constraint on stochastic forcing. Simulated examples illustrate the methodology.Comment: Published in SIAM Journal of Control and Optimizatio