Equality constraints bilinear

Remark 5 We can also treat the heat loads of each match as variables since they participate linearly in the energy balances. The penalty that we pay, however, is that the objective function no longer satisfies the property of convexity, and hence we will have two possible sources of nonconvexities the objective function and the bilinear equality constraints. 

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. 

For the bilinear balances the objective remains the same but the equality constraints are now defined by equation (17). We may develop these equations in a Taylor series in a neighborhood of xQ, which denotes a first approximation of the true values... 

Lee S. and Grossmann l.E. 2003. Global optimization of nonlinear generahzed disjunctive programming with bilinear equality constraints Applications to process networks, Comput. Chem. Eng., 27, 1557-1575. 

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. 

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