Quasi-convex

This section presents the definitions, properties and relationships of quasi-convex, quasi-concave, pseudo-convex and pseudo-concave functions. 

Definition 2.3.1 (Quasi-convex function) f(x) is quasi-convex if... 

Note that /(jc) is quasi-concave if —/(jc) is quasi-convex. 

Illustration 2.3.1 Figure 2.12 shows a quasi-convex and quasi-concave function. 

Remark 1 A convex function is also quasi-convex since... 

Figure 2.12 Quasi-convex and quasi-concave functions...

Note that f(x) is strictly quasi-concave if -/( ) is strictly quasi-convex. 

Illustration 2.3.2 Figure 2.13 shows a strictly quasi-convex and strictly quasi-concave function. Theorem 2.3.2... 

Let f(x) be a lower (upper) semicontinuousfunction on the convex set S in J n. If f(x) is strictly quasi-convex (strictly quasi-concave) on S, then... 

Properties of Quasi-convex and Quasi-concave Functions... 

Remark 1 Note that the summation of quasi-convex functions is not necessarily a quasi-convex function as in convex functions. Also, note that the summation of convex and quasi-convex functions is not necessarily a convex or quasi-convex function. 

Remark 2 Convex and concave functions do not satisfy properties (ii) and (iii) of the quasi-convex and quasi-concave functions. For instance, it is true that the reciprocal of a positive concave function is convex, but the reverse does not hold. As an example consider the function f(x) = ex which is convex and whose reciprocal is also convex. 

Relationships among Convex, Quasi-convex and Pseudo-convex Functions... 

The relationships among convex, quasi-convex and pseudo-convex functions are summarized in the following ... 

Section 2.3 focuses on the generalizations of convex and concave functions and treats the quasi-convex, quasi-concave, pseudo-convex and pseudo-concave functions, and their properties. Further reading in this subject is the excellent book of Avriel et al. (1988). 

Remark 2 Note that condition C1 of the OA/ER involves additional conditions of quasi-convexity in x for... 

Note though that the additional condition of Quasi-convexity of Tkh(x) needs to be imposed so as to obtain equivalence between the original model and the one with the relaxed set of equalities into inequalities. 

Therefore, the condition of quasi-convexity of Tkh(x) has to be checked at every iteration in the OA/ER algorithm, which is not needed in the GBD algorithm. 

In the previous section we showed that if convexity holds along with quasi-convexity of Tkh(x) < 0 and the constraint qualification, then the OA/ER terminates in fewer iterations than the v2-GBD since the lower bound of the OA/ER is better than the one of the v2-GBD. 

It should be noted, however, that we need to check whether Tkh(x) < 0 is quasi-convex at each iteration. If the quasi-convexity condition is not satisfied, then the obtained lower bound by OA/ER may not be a valid one that is, it may be above the global solution of the MINLP model. This may happen due to the potential invalid linearizations which may cut off part of the feasible region. 

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