Using convex nonlinear relaxations in the global optimization of nonconvex generalized disjunctive programs

Abstract

In this paper we present a framework to generate tight convex relaxations for nonconvex generalized disjunctive programs. The proposed methodology builds on our recent work on bilinear and concave generalized disjunctive programs for which strong linear relaxations can be generated and extends its application by allowing nonlinear relaxations. This is particular important for those cases in which the convex envelopes of the nonconvex functions arising in the formulations are nonlinear (e.g. fractional terms). This extension is now possible by using the latest developments in convex disjunctive programming. We test the performance of the method in three typical process systems engineering problems, namely, the optimization of process networks, reactor networks and heat exchanger networks.

Review of Mixed‐Integer Nonlinear and Generalized Disjunctive Programming Methods

“…Consider Grossmann [79,80] show the improvement in bound tightening of using basic steps in several instances, as summarized in Table 3. It is easy to see from Table 3 that the bound tightening technique can be greatly improved in some instances by using basic steps.…”

“…The authors solve the GDP model using a modified version of the logic based Outer Approximation, showing the efficiency of the algorithm in finding a global solutions fast. Ruiz-Femenia et al [80] present a logic based Outer Approximation algorithm for solving optimal control problems.…”

“…Later, Yeomans and Grossmann [95] formulated the two fundamental superstructures for process systems (state task network and state equipment network) with GDP. GDP methods have shown to improve solution times for linear, convex and nonconvex process flowsheet synthesis problems [55,80,82,95,96].…”

“…Ruiz and Grossmann [79,80] propose improving the Optimality-Based Bound Tightening technique by strengthening the continuous relaxation of the convex approximation. This is achieved by using a logic operation called basic step [81], which is the intersecting between two disjunctions.…”

A global Optimization Approach to Non-Convex Problems

“…The methods or algorithms for solving the nonconvex function are presented in the previous literatures, which may be classified into two categories: probabilistic and deterministic. The deterministic approach is effective and efficient on the condition that initial guess is near the global minimum, such as multidimensional geometric method [1], tight convex relaxations [2], parametric linearization relaxation algorithm [3], Global Barrier Exclusion algorithm [4]. The probabilistic methods are genetic algorithms [5], tunneling method [6] and auxiliary function approach [7], beta algorithm [8], heuristic Kalman algorithm [9], Which may find the optimal answer at the expense of requiring a high computation burden and offering a probabilistic convergence as well as be sensitive to the variation of the parameters.…”