Second-stage value function

This is the basic idea of a two-stage stochastic program with recoiurse. At the first stage, before a realization of the random variables first-stage decision variables X to optimize the expected value g x) = t[G x, >)] of an objective fimction G(x, to) that depends on the optimal second stage objective function. 

The result is a deterministic program, where the original second-stage decisions are not a function of the realized scenario, i.e., it is assumed the there is a single scenario problem and all decisions xEV and yEV have to be made before the observation. The corresponding optimization problem is called the expected value problem, (TV problem) and can be written as follows ... 

When the second stage decisions are real-valued variables, the value function Qu(x) is piecewise-linear and convex in x. However, when some of the second stage variables are integer-valued, the convexity property is lost. The value function Qafx) is in general non-convex and non-differentiable in x. The latter property prohibits the use of gradient-based search methods for solving (MASTER). 

A full theoretical treatment of Pgl within the Born-Oppenheimer approximation involves two stages. First, the potentials V (R), V+(R), and the width T(f ) of the initial state must be calculated in electronic-structure calculations at different values of the internuclear distance R—within the range relevant for an actual collision at a certain collision energy—and then, in the second stage the quantities observable in an experiment (e.g., cross sections, energy and angular distributions of electrons) must be calculated from the functions V+(R), V+(R), and T(f ), taking into account the dynamics of the heavy-particle collisions. 

The first HTU term contains the physical and fluid-dynamic parameter and the second NTU term expresses the number of theoretical stages as function of the solute concentration difference. The extractor-specific HTU value is, on the one hand, described by the quotient of flow rate and cross-sectional area of the column, and, on the other hand, it is characterised by the interfacial area per unit volume and the mass transfer coefficient. The former is mainly influenced by drop size and phase hold-up, the latter by the relative movement of the dispersed phase. These characteristic HTU values can be experimentally measured for a certain extractor type and are used for comparison with other extractors or for the projection of larger units. 

Let us observe that in both examples 1 and 2 the function G(% >) is piecewise linear for any realization of >, and hence is not everywhere differentiable. The same holds for the optimal value function Q(-, p) of the second-stage problem (15). If the distribution of the corresponding random variables is discrete, then the resulting expected value function is also piecewise hnear and hence is not everywhere differentiable. 

As an example consider the optimal value function of the second stage problem (15). Suppose that only the right-hand-side vector h = h(second-stage problem is random. Then Q(x, h) = G(h — Tx), where G(x) = min fy.Wy = y a 0). Suppose that the random vector h has a pdf /( ). By using the transformation z = h — Tx, we, obtain... 

The data presented in Table 9.3 show that particles with lower surface polarities (ATP) cause the polymer dissolved in the second stage monomer to have relatively higher interfacial tension values (y iyn)./aq.phase) against the aqueous phase. The PEMA (or PS) latexes produced using AffiN initiator, which contains only weak polar surface groups, exhibited low surface polarities and higher interfacial tensions as compared to both Dow PS seed particles. These results showed that the surface polarity of the particles is primarily influenced by the functional groups attached to the polymer chain ends, rather than the bulk hydrophilicity of the monomers employed. 

The dissolution of a gas in a liquid is basically a two-stage process. First a cavity must be formed within the solvent of sufficient size to accommodate the solute molecule, and this step is followed by the insertion of the solute within the solvent cavity with a resultant change in the energy of the system which depends primarily on the magnitude of the solvent-solute interactions. This second stage of the process is now reasonably well understood in terms of the statistical mechanical theories of fluid mixtures discussed briefly in previous sections, although accurate predictions of thermodynamic excess functions are only possible in a limited number of cases when auxiliary measurements enable explicit values of and x to be estimated. It is, as yet, not possible to predict gas solubilities with comparable confidence. 

This relationship is described in Fig. 9.4-4 where the ratio T./Tq is a function of the ratio Y2/YQ. Parameter is the exponent B of the Fieundlich equation. Note that a small value of B denotes an equilibrium favorable for adsorption. In this case the reduction of the ratio Y /Yg or the loading 7, for a given feed loading 7q is considerable in comparison to the second stage. Things are quite different for great exponents B which represent equilibria unfavorable for adsorption. 

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