Theory and Applications of Robust Optimization

In this paper we survey the primary research, both theoretical and applied, in the area of Robust Optimization (RO). Our focus is on the computational attractiveness of RO approaches, as well as the modeling power and broad applicability of the methodology. In addition to surveying prominent theoretical results of RO, we also present some recent results linking RO to adaptable models for multi-stage decision-making problems. Finally, we highlight applications of RO across a wide spectrum of domains, including finance, statistics, learning, and various areas of engineering.Comment: 50 page

Motion Planning of Uncertain Ordinary Differential Equation Systems

This work presents a novel motion planning framework, rooted in nonlinear programming theory, that treats uncertain fully and under-actuated dynamical systems described by ordinary differential equations. Uncertainty in multibody dynamical systems comes from various sources, such as: system parameters, initial conditions, sensor and actuator noise, and external forcing. Treatment of uncertainty in design is of paramount practical importance because all real-life systems are affected by it, and poor robustness and suboptimal performance result if it’s not accounted for in a given design. In this work uncertainties are modeled using Generalized Polynomial Chaos and are solved quantitatively using a least-square collocation method. The computational efficiency of this approach enables the inclusion of uncertainty statistics in the nonlinear programming optimization process. As such, the proposed framework allows the user to pose, and answer, new design questions related to uncertain dynamical systems. Specifically, the new framework is explained in the context of forward, inverse, and hybrid dynamics formulations. The forward dynamics formulation, applicable to both fully and under-actuated systems, prescribes deterministic actuator inputs which yield uncertain state trajectories. The inverse dynamics formulation is the dual to the forward dynamic, and is only applicable to fully-actuated systems; deterministic state trajectories are prescribed and yield uncertain actuator inputs. The inverse dynamics formulation is more computationally efficient as it requires only algebraic evaluations and completely avoids numerical integration. Finally, the hybrid dynamics formulation is applicable to under-actuated systems where it leverages the benefits of inverse dynamics for actuated joints and forward dynamics for unactuated joints; it prescribes actuated state and unactuated input trajectories which yield uncertain unactuated states and actuated inputs. The benefits of the ability to quantify uncertainty when planning the motion of multibody dynamic systems are illustrated through several case-studies. The resulting designs determine optimal motion plans—subject to deterministic and statistical constraints—for all possible systems within the probability space

Chance Constrained Mixed Integer Program: Bilinear and Linear Formulations, and Benders Decomposition

In this paper, we study chance constrained mixed integer program with consideration of recourse decisions and their incurred cost, developed on a finite discrete scenario set. Through studying a non-traditional bilinear mixed integer formulation, we derive its linear counterparts and show that they could be stronger than existing linear formulations. We also develop a variant of Jensen's inequality that extends the one for stochastic program. To solve this challenging problem, we present a variant of Benders decomposition method in bilinear form, which actually provides an easy-to-use algorithm framework for further improvements, along with a few enhancement strategies based on structural properties or Jensen's inequality. Computational study shows that the presented Benders decomposition method, jointly with appropriate enhancement techniques, outperforms a commercial solver by an order of magnitude on solving chance constrained program or detecting its infeasibility

Software tools for stochastic programming: A Stochastic Programming Integrated Environment (SPInE)

SP models combine the paradigm of dynamic linear programming with modelling of random parameters, providing optimal decisions which hedge against future uncertainties. Advances in hardware as well as software techniques and solution methods have made SP a viable optimisation tool. We identify a growing need for modelling systems which support the creation and investigation of SP problems. Our SPInE system integrates a number of components which include a flexible modelling tool (based on stochastic extensions of the algebraic modelling languages AMPL and MPL), stochastic solvers, as well as special purpose scenario generators and database tools. We introduce an asset/liability management model and illustrate how SPInE can be used to create and process this model as a multistage SP application