Optimization general formulation

Optimizing the formulation of micellar surfactant solutions used for enhanced oil recovery consists of obtaining interfacial tensions as low as possible in multiphase systems, which can be achieved by mixing the injected solution with formation fluids. The solubilization of hydrocarbons by the micellar phases of such systems is linked directly to the interfacial efficiency of surfactants. Numerous research projects have shown that the amount of hydrocarbons solubilized by the surfactant is generally as great as the interfacial tension between the micellar phase and the hydrocarbons. The solubilization of crude oils depends strongly on their chemical composition [155]. 

The ingredients of formulating optimization problems include a mathematical model of the system, an objective function that quantifies a criterion to be extremized, variables that can serve as decisions, and, optionally, inequality constraints on the system. When represented in algebraic form, the general formulation of discrete/continu-ous optimization problems can be written as the following mixed integer optimization problem ... 

The optimal operation of mixed production facilities - A general formulation and some approaches for the solution. Comput. Chem. Eng., 20, 897-904. 

The generalized valence bond (GVB) method was the earliest important generalization of the Coulson—Fischer idea to polyatomic molecules (13,14). The method uses OEOs that are free to delocalize over the whole molecule during orbital optimization. Despite its general formulation, the GVB method is usually used in its restricted form, referred to as GVB SOPP, which introduces two simplifications. The first one is the perfect-pairing (PP) approximation, in which only one VB structure is generated in the calculation. The wave function may then be expressed in the simple form of Equation 9.1, as a product of so-called geminal two-electron functions ... 

Mujtaba and Macchietto (1996) presented a more general formulation for optimal design and operation, dealing in particular with multiple separation duties, multicomponent mixtures, more complex operations (involving off-cuts) and more general objective functions. The method utilises a dynamic model (Type IV, Chapter 4) of the column in the form of a generic system of DAEs. Models of various rigor (type III and V, etc. of Chapter 4) can therefore be used. 

The third maj or class of search methods are genetic algorithms (GAs), which are widely used for docking purposes. GAs are stochastic optimization methods inspired by the concepts of evolution (172-174). The optimization problem is generally formulated in the Ian-... 

The updated model can be used to determine the optimum values for the optimization variables. The general formulation of the economic optimization problem is ... 

As far as the general formulation is concerned, it is interesting to note that there exist similarities between monotonically convergent algorithms for optimal control and local control methods. This relationship has only recently been elucidated by Salomon and Turinici [76]. 

Related to preparative chromatography, diverse objective functions appear. For instance, they may focus on the economics of new plant for a certain product, or on maximizing the productivity VSP as a function of the injection volume under the constraints of minimal purities and yields. In the latter case a general formulation of the optimization task can be... 

Note that there are eight heat-capacity flow rates, four temperatures, and nine equahty constraints, and hence, three decision variables (e.g., fi, F, and Fg). Also, Ar ,i is set at 10 K. In a more general formulation, it could be adjusted as an optimization variable. 

It is important to monitor photopolymerization reactions as they provide valuable information about the reaction kinetics, which can help optimize photocurable formulations to achieve desired properties, such as structural, physical, and mechanical, in the cured materials. This section provides a detailed summary of the available characterization techniques commonly used to investigate photopolymerization reactions. The parameters that are generally investigated are the rate of polymerization Rp), DC and the amount of RCs, induction period (ti), quantum yield of polymerization, kinetic chain lengths, and gelation time. 

There are other examples of efforts to eliminate or reduce solvents but the general trend is towards optimization of formulations to perform well in a specialized application. 

Unlike optimization models with a single objective function, the interest is on finding a set of solutions that describe how the improvement of a single objective function value impacts the value of the other objectives. This set is commonly known as the Pareto-optimal set and each of its elements as a Pareto optimal solution. In this respect, a general formulation of a multi-objective problem is ... 

Myerson [65] first introduced the problem of optimal mechanism design, in the context of an auction for a single item with a seller that seeks to maximize her expected revenue. We will provide a general formulation of the optimal mechanism design problem, to parallel the formulation of the efficient mechanism design problem. However, analytic solutions to the optimal mechanism design problem are known only for special cases. 

Optimal Multiattribute Auctions. Che [25] has proposed optimal multiattribute auctions for the special case of seller cost functions that are defined in terms of a single unknown parameter. Che s auctions are direct-revelation mechanisms, and he considers both first-price and second-price variations. The second-price variation is exactly the one-sided VCG mechanism, and Che is able to derive the optimal scoring function (or reported cost function) that the buyer should state to maximize her payoff in equilibrium. Branco [21] extends the analysis to consider the case of correlated seller cost functions. No optimal multiattribute auction is known for a more general formulation of the problem, for example for the case of preferential-independence. 

Note also that the last example in the chapter essentially entails determining the placement of safety inventory in the supply chain (since the reorder point specifies inventory carried over and above the level required to meet expected demand over the replenishment lead time), albeit a simple, serial supply chain. Magnanti et al. (2006) present a general formulation of fhis problem, specifically a non-linear optimization formulation subject to linear constraints, to solve for the optimal placement of safety stocks in a general supply chain network. Magnanti et al. actually generalize the problem... 

A number of studies have been published in the past dealing with structural optimization of RC structures. One of the earliest studies on this subject is the work by Frangopol (1986) where the general formulation of the deterministic optimization problem is reviewed and some of the main features of two general purpose deterministic... 

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