Adaptive Stochastic Optimization Procedures

Abstract

Consider the following stochastic programming problem:

EquationSource % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeaabmqaaa % qaaiGac2gacaGGPbGaaiOBaiaadweacaWGMbWaaSbaaSqaaiaaicda % aeqaaOWaaeWaaeaacaWG4bGaaiilaiaadMhaaiaawIcacaGLPaaaae % aacaWGfbGaamOzamaaBaaaleaacaWGPbaabeaakmaabmaabaGaamiE % aiaacYcacaWG5baacaGLOaGaayzkaaGaeyizImQaaGimaiaacYcaca % aMf8UaamyAaiabg2da9iaaigdacaGGSaGaeSOjGSKaaiilaiaadMea % caGGSaaabaGaamiEaiabgIGiolaadseadaWgaaWcbaGaaGimaaqaba % GccqGHckcZtuuDJXwAK1uy0HMmaeHbfv3ySLgzG0uy0HgiuD3BaGqb % biab-1risjab-bW9UnaaCaaaleqabaGaamOBaaaakiaacYcacaaMf8 % UaamyEamaabmaabaGaam4DaaGaayjkaiaawMcaaiabgIGiolab-1ri % snaaCaaaleqabaGaamyCaaaakiaac6caaaaaaa!7052!<Equation ID="Equ1"><EquationNumber>1</EquationNumber><EquationSource Format="MATHTYPE"><![CDATA[ $$\begin{array}{*{20}{c}} {\min E{{f}_{0}}\left( {x,y} \right)} \hfill \\ {E{{f}_{i}}\left( {x,y} \right) \leqslant 0,\quad i = 1, \ldots ,I,} \hfill \\ {x \in {{D}_{0}} \subset \mathbb{R}{{}^{n}},\quad y\left( w \right) \in {{\mathbb{R}}^{q}}.} \hfill \\ \end{array}$$

(3.6.1)

In (3.6.1), x is a decision vector to be selected from a closed, bounded subset D ₀ of the n-dimensional real Euclidean space ℝ ⁿ; y is a q-dimensional vector valued random variable; f _i , i = 0, 1,...,I, are respectively defined measurable functions; E is the symbol of mathematical expectation (the expected values are supposed to exist). Note that (3.6.1) is a fairly general stochastic programming model form; it encompasses—under suitable transformations—the ‘model block’ and types discussed in the previous chapter.