Feasible domain

Remark 1 The constraints that are active at a feasible points restrict the feasibility domain while the inactive constraints do not impose any restrictions on the feasibility in the neighborhood of S, defined as a ball of radius e around S, Be(x). 

In Appendix A of Aggarwal and Floudas (1990), a procedure is presented that calculates upper bounds on the flow rates of the overall by-pass streams. These bounds are useful in restricting the feasible domain of the remaining flow rates. 

Remark 1 The resulting optimization model is an MINLP problem. The objective function is linear for this illustrative example (note that it can be nonlinear in the general case) and does not involve any binary variables. Constraints (i), (v), and (vi) are linear in the continuous variables and the binary variables participate separably and linearly in (vi). Constraints (ii), (iii), and (iv) are nonlinear and take the form of bilinear equalities for (ii) and (iii), while (iv) can take any nonlinear form dictated by the reaction rates. If we have first-order reaction, then (iv) has bilinear terms. Trilinear terms will appear for second-order kinetics. Due to this type of nonlinear equality constraints, the feasible domain is nonconvex, and hence the solution of the above formulation will be regarded as a local optimum. 

The solution moves along the boundary of the feasible domain. 

Provided that the objective and constraint functions in (1) and (3) are continuous and the feasible domains of these problems are nonempty and bounded, optimal solution points Tip and it are guaranteed to exist for (1) and (3), respectively [2], Note that... 

Rule 4 If a vertex falls outside the boundaries of the feasible domain, an artihcially worst response should be assigned to it and one should proceed further with rules 1-3. This will force the simplex back into the boundaries. 

First three experiments (points 1,2, and 3) will be performed, according to the conditions dehned by the initial simplex (Si). By applying rule 1, the vertex with the worst response (point 1) is rejected and reflected to create point 4. Points 2, 3, and 4 then form the new simplex (S2). An experiment is then run at the conditions defined by point 4, and the procedure is repeated. For sim-plexes 2-7, all defined according to rule 1, the new experiment always yielded better results than at least one of the two remaining experiments of the preceding simplex. From simplex 7, point 7 is considered to be the worst, rejected, and reflected to point 10 (Sg). However, this vertex falls outside the boundaries of the feasible domain, and an undesirable response is assigned to point 10. 

The results of the studies on separating minimum and maximum boiling azeotropes with light, intermediate, or heavy entrainers are compared according to their operation steps and feasibility domains. The decisive property for designing an effective BED process, separating azeotropes, is the relative position of the entrainer to the azeotrope in the bubble point series. But the type of the azeotrope (minimum or maximum) can modify the existence of some limiting parameters (FA a , N,., >.,). 

The general formnlation of the problem of optimization of a material is the same as in the strnctnral optimization. An optimal material is described by a set of decisive variables x (i = 1,2,..., n) which minimize or maximize an optimization criterion or multiple criteria. The solutions should be within or on the border of the feasible domain. 

In Eq. 1,0 and [a, b represent the definition of an interval parameter where a and b are the lower and upper bounds of the interval parameter, respectively. Furthermore, A() and N denote the nominal value of an interval parameter, half the varied range of the interval parameter, and the number of interval parameters, respectively. When the uncertainty of structural parameters is described by the interval vector, it means that the feasible domain of interval parameters is constrained into an W-dimensional rectangle. 

In the info-gap model (Ben-Haim 2001), the level (or degree) of uncertainty is defined by a single uncertain parameter a. Based on the definition of an uncertain parameter a in the info-gap model, the feasible domain of the interval parameter X can be represented by an uncertainty set X (a) e R which is described by... 

Figure 3 illustrates the relationship between the robustness function and the feasible domain of stmctural design to satisfy the performance criterion / < / for two-dimensional interval parameters. The robustness function a is derived as the worst case of the objective function, i.e., the upper bound of the objective function / in U X , (x). However, when the number of the combinations of uncertain parameters is extremely large, it may be hard to evaluate the worst case of the objective function reliably. For this reason, an efficient uncertainty analysis method is desired which can evaluate the upper bound of... 

Figure 6 shows illustrative representation of the concept of info-gap robustness function. When the uncertainty level is 0C2, the response domain is just encircled by the permissible (or feasible) domain. In this case, the info-gap robustness function is given by a.2. 

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