Sampling, Risk Measure And Distributional Robustness For Uncertainties In Stochastic Optimization

Stochastic optimization is an important branch of mathematical programming. It isthe mathematics for decision making under uncertainty. As the combination of optimiza-tion theorem, probability and statistic, stochastic optimization has wide applications ina lot of areas, such as portfolio optimization, telecommunication, engineering and sup-ply chain. As the development of the society and the scientific and technological level,the uncertainty has more undetectable efect in daily life. In recent tens years, as an ef-fective method for making decision under uncertainty, stochastic optimization becomesmore and more important. In this thesis, we from three diferent but closely connect-ed views: sampling, risk measure and distributional robustness, to study the uncertaintyin stochastic optimization. Specially, we investigate the foundational probability theorywhich is the fundament of the stochastic optimization; the asymptotic convergence analy-sis of diferent stochastic optimization models, such as Conditional Value at Risk (CVaR)constrained programs and stochastic programs with second order stochastic dominanceconstraints, when sample average approximation (SAA) method is applied; the efectivealgorithms for stochastic programs with second order stochastic dominance constraintsand its applications in portfolio optimization and supply chain problem; and the asymp-totic convergence analysis of distributional robust optimization.The main contents and innovation are as follows:(1) We propose a new concept: almost H-calmness. We relax the key conditionof the classic uniform large deviation theorem by apply the new concept. The new u-niform large deviation theorem can be applied to establish uniform exponential rate ofconvergence of the Clarke subdiferential of a piecewise smooth random function whichis fundamental to many nonsmooth stochastic programming problems. This theorem isan important theoretical tool when we consider the asymptotic convergence analysis ofmany nonsmooth stochastic programming problems, such as CVaR constrained programsand stochastic programs with second order stochastic dominance constraints.(2) When SAA method is applied to some stochastic programs with risk measureconstraints, such as CVaR constrained programs and stochastic programs with second or-der stochastic dominance constraints, we give the detailed asymptotic convergence anal-ysis. We investigate the constraints qualification (CQ) of sample average approximation problem when the CQ is hold for original stochastic program; by applying uniform law oflarge numbers and the uniform large deviation theorem extended in this thesis, under somemoderate conditions, we prove that optimal solutions and stationary points obtained fromsolving sample average approximated problems converge with probability one (w.p.1) totheir true counterparts and the rate of convergence is exponential rate.(3) We propose an alternative way to solve the problem that stochastic programswith second order stochastic dominance constraints can not satisfy slater constraint qual-ification and then develop the exact penalization schemes for this model. Based on theexact penalization formulations, we apply a well known level function method in nons-mooth optimization to the penalized problems. We also propose a modified cutting-planemethod to solve the problem. We have applied the algorithms to a portfolio problem and asupply chain problem and carried out extensive numerical tests on our proposed methodsin comparison with the pervious algorithm. The numerical results show that our proposedmethods are more efcient.(4) For distributional robust optimization, we focus on the asymptotic convergenceanalysis of case when the approximate distributional set constructed through samples con-verges to the true distributional set. This allows us to apply the convergence results to anydistributional robust model when the distributional set satisfies some moderate condition-s. We prove that the conditions are satisfied when convergence of distributional sets areconstructed through moments, mixture distribution, and moments and covariance matrix.That means, by applying the convergence results established in section6, we can give thedetailed asymptotic convergence analysis of these models. In the case when a distribu-tional set is defined through moment conditions, we derive a Hofman type error boundfor a probabilistic system of inequalities and equalities and use it to establish a linearbound for the distance of two distributional sets under total variation metric.