Nonconvex nonlinear model

Constraints (4.1), (4.2), (4.3), (4.4), (4.5) and (4.6) constitute a nonconvex nonlinear model due to constraints (4.3) and (4.4), which involve bilinear terms. Nonconvexity, and not necessarily nonlinearity, is a disadvantageous feature in any model, since global optimality cannot be guaranteed. Therefore, if can be avoided, it should. This is achieved by either linearizing the model or using convexification techniques where applicable. In this instance, the first option was proven possible as shown below. 

Both the mixing process and the approximation of the product profiles establish nonconvex nonlinearities. The inclusion of these nonlinearities in the model leads to a relatively precise determination of the product profiles but do not affect the feasibility of the production schedules. A linear representation of both equations will decrease the precision of the objective but it will also eliminate the nonlinearities yielding a mixed-integer linear programming model which is expected to be less expensive to solve. 

Remark 1 The main motivation behind the development of the simplified superstructure was to end up with a mathematical model that features only linear constraints while the nonlinearities appear only in the objective function. Yee et al. (1990a) identified the assumption of isothermal mixing which eliminates the need for the energy balances, which are the nonconvex, nonlinear equality constraints, and which at the same time reduces the size of the mathematical model. These benefits of the isothermal mixing assumption are, however, accompanied by the drawback of eliminating from consideration a number of HEN structures. Nevertheless, as has been illustrated by Yee and Grossmann (1990), despite this simplification, good HEN structures can be obtained. 

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. 

In this overview we focus on the elastodynamical aspects of the transformation and intentionally exclude phase changes controlled by diffusion of heat or constituent. To emphasize ideas we use a one dimensional model which reduces to a nonlinear wave equation. Following Ericksen (1975) and James (1980), we interpret the behavior of transforming material as associated with the nonconvexity of elastic energy and demonstrate that a simplest initial value problem for the wave equation with a non-monotone stress-strain relation exhibits massive failure of uniqueness associated with the phenomena of nucleation and growth. 

We recall that our wave equation represents a long wave approximation to the behavior of a structured media (atomic lattice, periodically layered composite, bar of finite thickness), and does not contain information about the processes at small scales which are effectively homogenized out. When the model at the microlevel is nonlinear, one expects essential interaction between different scales which in turn complicates any universal homogenization procedure. In this case, the macro model is often formulated on the basis of some phenomenological constitutive hypotheses nonlinear elasticity with nonconvex energy is a theory of this type. 

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. 

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. 

Nonlinear problems— NLP and MINLP—can be further classified as convex or nonconvex, depending on the convexity of the objective function and feasible region. The understanding of the type of problem in terms of classification and convexity is very important in the utilization of modeling systems, since there are specific solvers and solution techniques for each type of problem, and depending on the problem, there may be local and global solutions. A more comprehensive study on mathanati-cal programming topics and continuous nonlinear optimization is out of the scope of this chapter. The interested reader is directed to references [4,6,7]. 

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