Goal driven optimization

Abstract

Achieving a target objective, goal or aspiration level are relevant aspects of decision making under uncertainties. We develop a goal driven stochastic optimization model that takes into account an aspiration level. Our model maximizes the shortfall aspiration level criterion, which encompasses the probability of success in achieving the goal and an expected level of under-performance or shortfall.The key advantage of the proposed model is its tractability. We show that proposed model is reduced to solving a small collection of stochastic linear optimization problems with objectives evaluated under the popular conditional value-at-risk (CVaR) measure. Using techniques in robust optimization, we propose a decision rule based deterministic approximation of the goal driven optimization problem by solving a polynomial number of subproblems, with each subproblem being a second order cone problem (SOCP).As an extension, we consider the probabilistic constrained problem where a system of linear inequalities with stochastic entries is required to remain feasible with high probability. We review SOCP approximations for the individual probabilistic constrained problem. Moreover, a new formulation is proposed for approximating joint probabilistic constrained problem. Improvement of the new method upon the standard approach is shown.We apply the goal driven model to project management and inventory planning problems and show experimentally that the proposed algorithms are computationally efficient.