arXiv subjects

Sixiang Zhao

Publications and source records attributed to Sixiang Zhao.

5 recordsLinked to original sources

arXiv2020

An Accelerated Fitted Value Iteration Algorithm for MDPs with Finite and Vector-Valued Action Space

This paper studies an accelerated fitted value iteration (FVI) algorithm to solve high-dimensional Markov decision processes (MDPs). FVI is an approximate dynamic programming algorithm that has desirable theoretical properties. However, it can be intractable when the action space is finite but vector-valued. To solve such MDPs via FVI, we first approximate the value functions by a two-layer neural network (NN) with rectified linear units (ReLU) being activation functions. We then verify that such approximators are strong enough for the MDP. To speed up the FVI, we recast the action selection problem as a two-stage stochastic programming problem, where the resulting recourse function comes from the two-layer NN. Then, the action selection problem is solved with a specialized multi-cut decomposition algorithm. More specifically, we design valid cuts by exploiting the structure of the approximated value functions to update the actions. We prove that the decomposition can find the global optimal solution in a finite number of iterations and the overall accelerated FVI is consistent. Finally, we verify the performance of the FVI algorithm via a multi-facility capacity investment problem (MCIP). A comprehensive numerical study is implemented, where the results show that the FVI is significantly accelerated without sacrificing too much in precision.

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arXiv2020

A Randomized Nonlinear Rescaling Method in Large-Scale Constrained Convex Optimization

We propose a new randomized algorithm for solving convex optimization problems that have a large number of constraints (with high probability). Existing methods like interior-point or Newton-type algorithms are hard to apply to such problems because they have expensive computation and storage requirements for Hessians and matrix inversions. Our algorithm is based on nonlinear rescaling (NLR), which is a primal-dual-type algorithm by Griva and Polyak {[{Math. Program., 106(2):237-259, 2006}]}. NLR introduces an equivalent problem through a transformation of the constraint functions, minimizes the corresponding augmented Lagrangian for given dual variables, and then uses this minimizer to update the dual variables for the next iteration. The primal update at each iteration is the solution of an unconstrained finite sum minimization problem where the terms are weighted by the current dual variables. We use randomized first-order algorithms to do these primal updates, for which they are especially well suited. In particular, we use the scaled dual variables as the sampling distribution for each primal update, and we show that this distribution is the optimal one among all probability distributions. We conclude by demonstrating the favorable numerical performance of our algorithm.

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arXiv2019

A Flexible Multi-Facility Capacity Expansion Problem with Risk Aversion

This paper studies flexible multi-facility capacity expansion with risk aversion. In this setting, the decision maker can periodically expand the capacity of facilities given observations of uncertain demand. We model this situation as a multi-stage stochastic programming problem. We express risk aversion in this problem through conditional value-at-risk (CVaR), and we formulate a mean-CVaR objective. To solve the multi-stage problem, we optimize over decision rules. In particular, we approximate the full policy space of the problem with a tractable family of if-then policies. Subsequently, a decomposition algorithm is proposed to optimize the decision rule. This algorithm decomposes the model over scenarios and it updates solutions via the subgradients of the recourse function. We demonstrate that this algorithm can quickly converge to high-performance policies. To illustrate the practical effectiveness of this method, a case study on the waste-to-energy system in Singapore is presented. These simulation results show that by adjusting the weight factor of the objective function, decision makers are able to trade off between a risk-averse policy that has a higher expected cost but a lower value-at-risk, and a risk-neutral policy that has a lower expected cost but a higher value-at-risk risk.

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arXiv2019

An Inexact Primal-Dual Algorithm for Semi-Infinite Programming

This paper considers an inexact primal-dual algorithm for semi-infinite programming (SIP) for which it provides general error bounds. To implement the dual variable update, we create a new prox function for nonnegative measures which turns out to be a generalization of the Kullback-Leibler divergence for probability distributions. We show that under suitable conditions on the error, this algorithm achieves an $\mathcal{O}(1/\sqrt{K})$ rate of convergence in terms of the optimality gap and constraint violation. We then use our general error bounds to analyze the convergence and sample complexity of a specific primal-dual SIP algorithm based on Monte Carlo integration. Finally, we provide numerical experiments to demonstrate the performance of our algorithm.

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arXiv2018

Corporative Stochastic Approximation with Random Constraint Sampling for Semi-Infinite Programming

We developed a corporative stochastic approximation (CSA) type algorithm for semi-infinite programming (SIP), where the cut generation problem is solved inexactly. First, we provide general error bounds for inexact CSA. Then, we propose two specific random constraint sampling schemes to approximately solve the cut generation problem. When the objective and constraint functions are generally convex, we show that our randomized CSA algorithms achieve an $\mathcal{O}(1/\sqrt{N})$ rate of convergence in expectation (in terms of optimality gap as well as SIP constraint violation). When the objective and constraint functions are all strongly convex, this rate can be improved to $\mathcal{O}(1/N)$.

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