Region nonconvex

Now consider the influence of the inequality constraints on the optimization problem. The effect of inequality constraints is to reduce the size of the solution space that must be searched. However, the way in which the constraints bound the feasible region is important. Figure 3.10 illustrates the concept of convex and nonconvex regions. 

The addition of inequality constraints complicates the optimization. These inequality constraints can form convex or nonconvex regions. If the region is nonconvex, then this means that the search can be attracted to a local optimum, even if the objective function is convex in the case of a minimization problem or concave in the case of a maximization problem. In the case that a set of inequality constraints is linear, the resulting region is always convex. 

The solution 0C of max-min problem (16) is the critical point which limits the RI that is, it is the point where the largest inscribed polytope meets the feasible region R. In general, 8C need not correspond to a vertex of the polytope (e.g., for some nonconvex feasible regions R). However, to date no general algorithm has been developed to find nonvertex critical points which limit the RI. 

A sufficient condition that the RI be determined by a vertex critical point is that the feasible region R be convex. (Of course, a special case of this is when all the feasibility constraints are linear see Section III,B.) Unfortunately, when flow rates or heat transfer coefficients are included in the uncertainty range, the feasible region can be nonconvex (see Examples 1 and 2 and Section III,C,3). Thus, current algorithms for calculating the RI are limited to temperature uncertainties only. 

Fig. 9. Nonconvex feasible region for a class 2 problem (Examples 6 and 12).

The feasible region defined by these constraints is plotted in Fig. 13. The feasible region is nonconvex, and thus the corner point theorem does not hold. In particular, the HEN is not feasible for 422 K s rf s 508 K, even though it is feasible for the corner points of the uncertainty range rf = 415 K and rf = 515 K. 

The feasible region is depicted in Figure 6.4 and is nonconvex due to the bilinear inequality constraint. This problem exhibits a strong local minimum at (x, y) — (4,1) with objective equal to -5, and a global minimum at (x, y) = (0.5,8) with objective equal to -8.5. 

Skinny molecular range, [af, a< ) af is defined above, whereas is the maximum threshold at and below which all locally nonconvex domains on the surface of density domains are simply connected. In simpler terms, in the skinny molecular range all nuclei are found within a single density domain, but there are formal "neck regions on the surface of density domains. In the terminology of shape group analysis [2], rings of D) type can be found on the surface of density domains. 

If any two points in the feasible region can be found such that some point on a straight line between them lies outside the feasible region, then the feasible region is nonconvex, as illustrated in Figure 1.14b. 

The importance of convexity is that problems with a convex feasible region are more easily solved to a global optimum. Problems with nonconvex feasible regions are prone to convergence to local minima. 

All of the nonlinear programming algorithms can suffer from the convergence and local optima problems described in Section 1.9.7. Probabilistic methods such as simulated annealing and genetic algorithms can be used if it is suspected that the feasible region is nonconvex or multiple local optima are present. 

Corpulent molecular range No locally nonconvex multiply connected set on the surface of the density domain (no neck region) occurs, but there exists at least one local nonconvex region along the surface of the density domain DD (K,a). 

Moreover, the proposed procedure suffers from the same drawbacks as aU the mixed models of the scaled methods The boundaries of the GMR acceptance region converge to a minimum at the switching variability value and then start to spread apart for higher values of CV, presenting a nonconvex shape (Fig. 1). Consequently, an additional point estimate constraint criterion on GMR is needed. 

Figure 3.9 Property 1. The AR is convex, (a) A nonconvex region and (b) a convex region generated via mixing operations.

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