Nonconvexities MINLP

Vaidyanathan, R. and El-Halwagi, M. M. (1996). Global optimization of nonconvex MINLP s by interval analysis. In Global Optimization in Engineering Design, (I. E. Grossmann, ed.), pp. 175-194. Kluwer Academic Publishers, Dordrecht, The Netherlands. 

Scenario 4 Formulation for fixed water quantity with reusable water storage Constraints (4.18), (4.19), (4.3), (4.20), (4.16), (4.17), (4.21), (4.22), (4.23), (4.24), (4.25) and (4.26) together constitute a complete water reuse/recycle model for a situation in which the quantity of water in each water using operation is fixed. This is also a nonconvex MINLP for which exact linearization is not possible. 

As shown in Table 4.4, the model for scenario 2, which is a nonconvex MINLP, consists of 1195 constraints, 352 continuous and 70 binary variables. An average of 151 nodes were explored in the branch and bound algorithm over the 3 major iterations between the MILP master problem and NLP subproblems. The problem was solved in 2.48 CPU seconds with an objective value of 1.67 million. Whilst the product quantity is the same as in scenario 1, i.e. 850 t, the water requirement is only 185 t, which corresponds to 52.56% reduction in freshwater requirement. The water network to achieve this target is shown in Fig. 4.15. 

The foregoing constraints constitute the full heat storage model. With the exception of constraints (11.3)—(11.5), all the constraints are linear. Constraints (11.3)—(11.5) entail nonconvex bilinear terms which render the overall model a nonconvex MINLP. However, the type of bilinearity exhibited by these constraints can be readily removed without compromising the accuracy of the model using the so called Glover transformation, which has been used extensively in the foregoing chapters of this book. This is demonstrated underneath using constraints (11.3). 

The resulting optimization model is a nonconvex MINLP and the use of OA/ER/AP or v2-GBD can only provide a local optimum solution. In fact, application of v2-GBD or OA/ER/AP by projecting only on the binary variables showed that there exist several local solutions and the solution found depends heavily on the starting point. As a result, Aggarwal and Floudas (1990) proposed another projection scheme, which is... 

Table 9.1 shows how outer approximation, as implemented in the DICOPT software, performs when applied to the process selection model in Example 9.3. Note that this model does not satisfy the convexity assumptions because its equality constraints are nonlinear. Still DICOPT does find the optimal solution at iteration 3. Note, however, that the optimal MILP objective value at iteration 3 is 1.446, which is not an upper bound on the optimal MINLP value of 1.923 because the convexity conditions are violated. Hence the normal termination condition that the difference between upper and lower bounds be less than some tolerance cannot be used, and DICOPT may fail to find an optimal solution. Computational experience on nonconvex problems has shown that retaining the best feasible solution found thus far, and stopping when the objective value of the NLP subproblem fails to improve, often leads to an optimal solution. DICOPT stopped in this example because the NLP solution at iteration 4 is worse (lower) than that at iteration 3. 

Remark 4 The presented optimization model is an MINLP problem. The binary variables select the process stream matches, while the continuous variables represent the utility loads, the heat loads of the heat exchangers, the heat residuals, the flow rates and temperatures of the interconnecting streams in the hyperstructure, and the area of each exchanger. Note that by substituting the areas from the constraints (B) into the objective function we eliminate them from the variable set. The nonlinearities in the in the proposed model arise because of the objective function and the energy balances in the mixers and heat exchangers. As a result we have nonconvexities present in both the objective function and constraints. The solution of the MINLP model will provide simultaneously the... 

The objective function is nonlinear and nonconvex and hence despite the linear set of constraints the solution of the resulting optimization model is a local optimum. Note that the resulting model is of the MINLP type and can be solved with the algorithms described in the chapter of mixed-integer nonlinear optimization. Yee and Grossmann (1990) used the OA/ER/AP method to solve first the model and then they applied the NLP suboptimization problem for the fixed structure so as to determine the optimal flowrates of the split streams if these take place. 

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. 

Remark 1 The mathematical model is an MINLP problem since it has both continuous and binary variables and nonlinear objective function and constraints. The binary variables participate linearly in the objective and logical constraints. Constraints (i), (iv), (vii), and (viii) are linear while the remaining constraints are nonlinear. The nonlinearities in (ii), (iii), and (vi) are of the bilinear type and so are the nonlinearities in (v) due to having first-order reactions. The objective function also features bilinear and trilinear terms. As a result of these nonlinearities, the model is nonconvex and hence its solution will be regarded as a local optimum unless a global optimization algorithm is utilized. 

In MINLP, however, there is the additional complication that nonlinearities can often be formulated in many different, but equivalent, ways and, as expected. this can have a great impact on the performance of MINLP algorithms, particularly with respect to the nonconvexities of the nonlinear constraints. 

The MINLP model for the synthesis problem consists of minimizing the objective function in (29) subject to the feasible space defined by eqs. (21)-(28). The continuous variables (/. q, ( hu, qcu, dt, dtcu. discrete variables c, ecu, zhu are O-I. The advantage of this model is that the constraints (21)-(28) are all linear. The nonlinearities have all been placed in the objective function (29). However, it should be noted that since these terms are nonconvex, the MINLP may lead to local optimal solutions. 

Zero flows in process flowsheets Large size of MINLP problems Nonconvexities... 

Kocis, G. R., and Grossmann, I. E. Global Optimization of Nonconvex Mixed-Integer Nonlinear Programming (MINLP) Problems in Process Synthesis, Ind. Eng. Chem. Res. 27,1407 (1988). 

Ryoo H.S. and Sahinidis N.V. 1995. Global optimization of nonconvex NLPs and MINLPs with applications in process design, Comput. Chem. Eng., 19(5), 551-566. 

Kocis, G.R. and Grossmann, I.E. (1988) Global optimisation of nonconvex mixed-integer non linear programming (MINLP) problems in process synthesis. Industrial Engineering Chemistry Research, 27 (8), 1407-1421. 

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