Convex lower bounding

For the distillation problem, each equation of the MESH system is converted into two inequalities. It is required (Floudas and Maranas, 1995) to apply convex lower bounding to nonlinear h/x) and nonconvex gh(x) to get all solutions. After separating nonlinear equalities from hj(x) and nonconvex inequalities from gij(x), a convex relaxation can be written. For distillation, h/x) contains the MESH system of N(2C+3) equations, where N is number of stages and C is number of components. For a simulation problem, giJ(x) is empty but for design problems, it may contain inequality constraints. The liquid and vapor composition, stage temperatures, liquid and vapor flow rates, and heat loads of condenser and reboiler form vector x. 

An alternative approach based on convex lower bounding and partitioning was proposed by Maranas et al. [20] which is used in this paper and is presented in detail in Section 2. The simultaneous decoupling and pole placement conditions without the cancellation of invariant zeros are derived in Section 3. Three examples are then presented in Section 4 and finally the work is summarized in section 5. 

The approach proposed by Maranas et al. [20] is based on creating a convex lower bounding function coupled with a partitioning strategy and like Interval-Newton methods, it... 

Step 2 Replace the nonconvex functions by convex lower bounding functions that results in the following problem (R). 

A detailed description of the convex lower bounding function of special nonconvex functions can be found in [20]. For general twice differential nonconvex functions, the basic form of the convex lower bounding function L is the following ... 

Step 3 Solve the convex lower bounding problem using a local optimization algorithm (e.g. MINOS [21], NPSOL [22]) which provides a lower bound for the solution of the original problem. 

In the case of a biUnear term xy. Ref. 21 showed that the tightest convex lower bound over the domain x [/, > ] is obtained by introducing a new... 

The convex lower bounding function /(x), dehned over the rectangular domain of x < x < x, possesses a number of important properties that guarantee the convergence of the aBB algorithm to the global optimum solution ... 

The key development in the convex lower bounding formulation is the dehnition of the a parameters. Specihcally, the magnitude of the a parameters may be related to the minimum eigenvalue of the Hessian matrix of the nonconvex term/(x) ... 

The quality of the convex lower bounding problem can also be improved by ensuring that the variable bounds are as tight as possible. These variable bound updates can be performed either at the onset of an aBB run or at each iteration. 

Because the maximum separation between the nonconvex terms and their respective convex lower bounding functions is both a bounded and a continuous function of the size of rectangular domain, arbitrarily small feasibility and convergence tolerance limits are attained for a finite-sized partition element. 

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