Generalized Decision Rule Approximations for Stochastic Programming via Liftings

Stochastic programming provides a versatile framework for decision-making under uncertainty, but the resulting optimization problems can be computationally demanding. It has recently been shown that, primal and dual linear decision rule approximations can yield tractable upper and lower bounds on the optimal value of a stochastic program. Unfortunately, linear decision rules often provide crude approximations that result in loose bounds. To address this problem, we propose a lifting technique that maps a given stochastic program to an equivalent problem on a higherdimensional probability space. We prove that solving the lifted problem in primal and dual linear decision rules provides tighter bounds than those obtained from applying linear decision rules to the original problem. We also show that there is a one-to-one correspondence between linear decision rules in the lifted problem and families of non-linear decision rules in the original problem. Finally, we identify structured liftings that give rise to highly flexible piecewise linear decision rules and assess their performance in the context of a stylized investment planning problem.

Multiarmed Bandits Problem Under the Mean-Variance Setting

The classical multi-armed bandit (MAB) problem involves a learner and a collection of K independent arms, each with its own ex ante unknown independent reward distribution. At each one of a finite number of rounds, the learner selects one arm and receives new information. The learner often faces an exploration-exploitation dilemma: exploiting the current information by playing the arm with the highest estimated reward versus exploring all arms to gather more reward information. The design objective aims to maximize the expected cumulative reward over all rounds. However, such an objective does not account for a risk-reward tradeoff, which is often a fundamental precept in many areas of applications, most notably in finance and economics. In this paper, we build upon Sani et al. (2012) and extend the classical MAB problem to a mean-variance setting. Specifically, we relax the assumptions of independent arms and bounded rewards made in Sani et al. (2012) by considering sub-Gaussian arms. We introduce the Risk Aware Lower Confidence Bound (RALCB) algorithm to solve the problem, and study some of its properties. Finally, we perform a number of numerical simulations to demonstrate that, in both independent and dependent scenarios, our suggested approach performs better than the algorithm suggested by Sani et al. (2012)