Constrained Optimization Problems (COP’s) are encountered in many scientific fields concerned with industrial applications such as kinematics, chemical process optimization, molecular design, etc. When non-linear relationships among variables are defined by problem constraints resulting in non-convex feasible sets, the problem of identifying feasible solutions may become very hard. Consequently, finding the location of the global optimum in the COP is more difficult as compared to bound-constrained global optimization problems. This chapter proposes a new interval partitioning method for solving the COP. The proposed approach involves a new subdivision direction selection method as well as an adaptive search tree framework where nodes (boxes defining different variable domains) are explored using a restricted hybrid depth-first and best-first branching strategy. This hybrid approach is also used for activating local search in boxes with the aim of identifying different feasible stationary points. The proposed search tree management approach improves the convergence speed of the interval partitioning method that is also supported by the new parallel subdivision direction selection rule

Summary. A new efficient interval partitioning approach to solve constrained global optimization problems is proposed. This involves a new parallel subdivision direction selection method as well as an adaptive tree search. The latter explores nodes (intervals in variable domains) using a restricted hybrid depth-first and bestfirst branching strategy. This hybrid approach is also used for activating local search to identify feasible stationary points. The new tree search management technique results in improved performance across standard solution and computational indicators when compared to previously proposed techniques. On the other hand, the new parallel subdivision direction selection rule detects infeasible and suboptimal boxes earlier than existing rules, and this contributes to performance by enabling earlier reliable deletion of such subintervals from the search space. Key words: constrained global optimization, interval partitioning, adaptive search tree management, subdivision direction selection rules, parsing. 1

In deterministic global optimization algorithms for constrained problems, it can be advantageous to identify and utilize coordinates in which the problem is convex, as Epperly and Pistikopoulos have done. In self-validating versions of these algorithms, a useful technique is to construct regions about approximate optima, within which unique local optima are known to exist; these regions are to be as large as possible, for exclusion from the continuing search process. In this paper, we clarify the theory and develop algorithms for constructing such large regions, when we know the problem is convex in some of the variables. In addition, this paper clarifies how one can validate existence and uniqueness of local minima when using the Fritz John equations in the general case. We present numerical results that provide evidence of the efficacy of our techniques.