Panayiotis Lemonidis’s research while affiliated with Massachusetts Institute of Technology and other places

A bounding algorithm for the global solution of nonlinear bilevel programs involving nonconvex functions in both the inner and outer programs is presented. The algorithm is rigorous and terminates finitely to a point that satisfies ε-optimality in the inner and outer programs. For the lower bounding problem, a relaxed program, containing the constraints of the inner and outer programs augmented by a parametric upper bound to the parametric optimal solution function of the inner program, is solved to global optimality. The optional upper bounding problem is based on probing the solution obtained by the lower bounding procedure. For the case that the inner program satisfies a constraint qualification, an algorithmic heuristic for tighter lower bounds is presented based on the KKT necessary conditions of the inner program. The algorithm is extended to include branching, which is not required for convergence but has potential advantages. Two branching heuristics are described and analyzed. Convergence proofs are provided and numerical results for original test problems and for literature examples are presented.

A modeling framework based on bilevel programs involving nonconvex functions and mixed-integer variables is presented. Bilevel programs are programs where one optimization problem (upper-level program) is constrained by the (global) solutions of another optimization problem (lower-level program). Bilevel programs represent hierarchical decision making where each of two decision makers (optimizers) controls a subset of the decision variables. After discussing existing applications from the literature, new formulations from process design and operations are proposed, including parameter estimation in thermodynamics, scenario-based design and decision making in monopoly markets. In many literature applications and all the proposed problems, the lower-level program is nonconvex and therefore can only be solved rigorously based on recent algorithmic advances [1]. Bilevel programs have been used extensively in operations research for several decades. In this discipline typically the optimizers represent actual decision makers, such as policy makers, military leaders or industrial managers. For instance, Bracken and McGill [2] consider the allocation of military resources; the optimizers model the decision makers of two sides in a war. Fortuny-Amat and McCarl [3] discuss the case of irrigation during limited water supply; here the upper-level program corresponds to the government entity which allocates water to the various farmers; the lower-level programs represent the farmers which given an allocation try to optimize their production. The first application proposed in this talk is decision making by pharmaceutical companies which have monopoly or oligopoly. It is shown that this problem can be cast as a mixed-integer bilevel program with nonconvex objective functions. Often in operations research linear models are sufficient. However, this is not the case in process design. Clark and Westerberg [4] discuss multi-level and multi-objective optimization problems and show that design under thermodynamic equilibrium can be cast as a bilevel optimization problem. The upper-level optimizer is the designer, while the lower-level optimizer is nature which (at constant temperature and pressure) minimizes Gibbs free energy. Thus, the lower-level program is nonconvex and multi-modal, except for special cases such as ideal solutions [5]. Clark and Westerberg [4] recognize the difficulties of nonconvexity and introduce a variation of the original formulation where a solution is defined as a local solution; this change of what a solution means may be acceptable in some cases, but in many it is not. Another application of bilevel programs in process design is feasibility and flexibility analysis [6,7]; here the upper-level program is the designer, while the lower-level program is the maximization of constraint violation; here a global solution of the lower-level program is necessary, for otherwise an infeasible point may be obtained. Maranas and co-workers [8,9] consider optimal gene knockout; the upper-level program is the designer which tries to obtain maximal yield of a chemical from a microorganism, while the lower-level optimizer is the microorganism which will maximize its growth; for metabolic networks linear formulations are used, so nonconvexity is not an issue. An extension of feasibility-flexibility analysis is scenario-integrated dynamic optimization. In this talk formulations by Abel and Marquardt [10] are analyzed and extended, and solution methods are proposed. The goal of the formulations in this section is to account for uncertainty in the operation of dynamic systems. During process operation events outside the control of the operator can occur, such as drastic price changes, change of weather or failure of a process component. These events change the dynamics of the system. If these events are not taken into account during the plant design phase the operation may become infeasible and result in catastrophic events. The upper-level objective is to optimize the nominal operation, i.e., the operation without the occurrence of the external events. The lower-level problem is that the scenario operation, i.e., the operation once the event has occurred, is feasible. The global solution of the lower-level programs is mandated similar to feasibility and flexibility problems. A third application of bilevel programs is based on parameter estimation in thermodynamic equilibrium. In process design and operation typically the phase equilibria are calculated based on activity coefficient models such as the NRTL model. These models contain adjustable parameters which cannot be measured directly, but rather must be estimated based on measured phase-splits. In this talk it is shown that this parameter estimation naturally leads to a bilevel optimization problem. The upper-level program aims in minimizing the error in predictions by adjusting the model parameters, e.g., through a least squares error objective function. The lower-level programs represent the thermodynamic stability requirement, i.e., the predictions must be the global minimum of the Gibbs free energy for the chosen parameter values; equivalent stability criteria [11,12] can be used instead. The lower-level programs are nonconvex and their global solution is necessary. [1] A. Mitsos, P. Lemonidis, and P. I. Barton. Global solution of bilevel programs with a nonconvex inner program. In press: Journal of Global Optimization, October 12, 2007. [2] J. Bracken and J. T. McGill. Mathematical programs with optimization problems in constraints. Operations Research, 21(1):37-44, 1973. [3] J. Fortuny-Amat and B. McCarl. A representation and economic interpretation of a two-level programming problem. Journal of the Operational Research Society, 32(9):783-792, 1981. [4] P. A. Clark and A. W. Westerberg. Optimization for design problems having more than one objective. Computers & Chemical Engineering, 7(4):259-278, 1983. [5] W. Robert Smith and R. W. Missen. Chemical Reaction Equilibrium Analysis: Theory and Algorithms. John Wiley & Sons, New York, 1982. [6] I. E. Grossmann and K. P. Halemane. Decomposition strategy for designing flexible chemical-plants. AIChE Journal, 28(4):686-694, 1982. [7] C. A. Floudas, Z. H. Gms, and M. G. Ierapetritou. Global optimization in design under uncertainty: Feasibility test and flexibility index problems. Industrial & Engineering Chemistry Research, 40(20):4267-4282, 2001. [8] A. P. Burgard and C. D. Maranas. Optimization-based framework for inferring and testing hypothesized metabolic objective functions. Biotechnology and Bioengineering, 82(6):670-677, 2003. [9] A. P. Burgard, P. Pharkya, and C. D. Maranas. OptKnock: A bilevel programming framework for identifying gene knockout strategies for microbial strain optimization. Biotechnology and Bioengineering, 84(6):647-657, 03. [10] O. Abel and W. Marquardt. Scenario-integrated modeling and optimization of dynamic systems. AIChE Journal, 46(4):803-823, 2000. [11] L. E. Baker, A. C. Pierce, and K. D. Luks. Gibbs energy analysis of phase equilibria. Soc. Petrol. Engrs. J., 22:731-742, 1982. [12] A. Mitsos and P. I. Barton. A dual extremum principle in thermodynamics. AIChE Journal, 53(8):2131-2147, 2007.

The goals of this thesis are the development of global optimization algorithms for semi-infinite and generalized semi-infinite programs and the application of these algorithms to kinetic model reduction. The outstanding issue with semi-infinite programming (SIP) was a methodology that could provide a certificate of global optimality on finite termination for SIP with nonconvex functions participating. We have developed the first methodology that can generate guaranteed feasible points for SIP and provide e-global optimality on finite termination. The algorithm has been implemented in a branch-and-bound (B&B) framework and uses discretization coupled with convexification for the lower bounding problem and the interval constrained reformulation for the upper bounding problem. Within the framework of SIP we have also proposed a number of feasible-point methods that all rely on the same basic principle; the relaxation of the lower-level problem causes a restriction of the outer problem and vice versa. All these methodologies were tested using the Watson test set. It was concluded that the concave overestimation of the SIP constraint using McCormcick relaxations and a KKT treatment of the resulting expression is the most computationally expensive method but provides tighter bounds than the interval constrained reformulation or a concave overestimator of the SIP constraint followed by linearization. All methods can work very efficiently for small problems (1-3 parameters) but suffer from the drawback that in order to converge to the global solution value the parameter set needs to subdivided. Therefore, for problems with more than 4 parameters, intractable subproblems arise very high in the B&B tree and render global solution of the whole problem infeasible.

Finite formulations are presented for the calculation of lower and upper bounds on the optimal solution value of semi-infinite programs (SIPs) involving smooth, potentially nonconvex objective function and constraints. The lower bounding problem is obtained by a formulation that combines the first- and second-order KKT necessary conditions of the lower-level problem with a discretization of the parameter set. The resulting mathematical program with equilibrium constraints (MPEC) is a relaxation of the original SIP and furnishes valid lower bounds. If the parameter set is subdivided, the optimal solution value of the lower bounding problem converges to the optimal solution value of the SIP. The upper bounding problem is based on convex and linear relaxations of the lower-level problem resulting in a restriction of the SIP. If the parameter set is subdivided, the constructed relaxations converge to the original lower-level program. The existence of SIP Slater points ensures convergence of the upper bounding problems to the optimal solution value of the SIP. Several alternatives for the upper bounding problem are presented and discussed. Numerical results are presented for a number of test problems from the literature.

Optimization problems involving a finite number of decision variables and an infinite number of constraints are referred to as semi-infinite programs (SIPs). Existing numerical methods for solving nonlinear SIPs make strong assumptions on the properties of the feasible set, e.g., convexity and/or regularity, or solve a discretized approximation which only guarantees a lower bound to the true solution value of the SIP. Here, a general, deterministic algorithm for solving semi-infinite programs to guaranteed global optimality is presented. A branch-and-bound (B&B) framework is used to generate convergent sequences of upper and lower bounds on the SIP solution value. The upper-bounding problem is generated by replacing the infinite number of real-valued constraints with a finite number of constrained inclusion bounds; the lower-bounding problem is formulated as a convex relaxation of a discretized approximation to the SIP. The SIP B&B algorithm is shown to converge finitely to –optimality when the subdivision and discretization procedures used to formulate the node subproblems are known to retain certain convergence characteristics. Other than the properties assumed by globally-convergent B&B methods (for finitely-constrained NLPs), this SIP algorithm makes only one additional assumption: For every minimizer x* of the SIP there exists a sequence of Slater points xn for which (cf. Section 5.4). Numerical results for test problems in the SIP literature are presented. The exclusion test and a modified upper-bounding problem are also investigated as heuristic approaches for alleviating the computational cost of solving a nonlinear SIP to guaranteed global optimality.