Marco A. Duran

Bio: Marco A. Duran is an academic researcher from Carnegie Mellon University. The author has contributed to research in topics: Nonlinear programming & Optimization problem. The author has an hindex of 3, co-authored 4 publications receiving 1705 citations.

An outer-approximation algorithm for a class of mixed-integer nonlinear programs

Abstract: An outer-approximation algorithm is presented for solving mixed-integer nonlinear programming problems of a particular class. Linearity of the integer (or discrete) variables, and convexity of the nonlinear functions involving continuous variables are the main features in the underlying mathematical structure. Based on principles of decomposition, outer-approximation and relaxation, the proposed algorithm effectively exploits the structure of the problems, and consists of solving an alternating finite sequence of nonlinear programming subproblems and relaxed versions of a mixed-integer linear master program. Convergence and optimality properties of the algorithm are presented, as well as a general discussion on its implementation. Numerical results are reported for several example problems to illustrate the potential of the proposed algorithm for programs in the class addressed in this paper. Finally, a theoretical comparison with generalized Benders decomposition is presented on the lower bounds predicted by the relaxed master programs.

Simultaneous optimization and heat integration of chemical processes

Abstract: A procedure is proposed for simultaneously handling the problem of optimal heat integration while performing the optimization of process flow sheets. The method is based on including a set of constraints into the nonlinear process optimization problem so as to insure that the minimum utility target for heat recovery networks is featured. These heat integration constraints, which do not require temperature intervals for their definition, are based on a proposed representation for locating pinch points that can vary according to every set of process stream conditions (flow rates and temperatures) selected in the optimization path. The underlying mathematical formulations correspond to nondifferentiable optimization problems, and an efficient smooth approximation method is proposed for their solution. An example problem on a chemical process is presented to illustrate the economic savings that can be obtained with the proposed simultaneous approach. The method reduces to simple models for the case of fixed flow rates and temperatures.

A mixed-integer nonlinear programming algorithm for process systems synthesis

Abstract: The problem of synthesizing processing systems via simultaneous structural and parameter optimization is addressed in this paper. Based on a superstructure representation for embedding alternative configurations, a general mixed-integer nonlinear programming (MINLP) framework is presented for the synthesis problem. An efficient outer-approximation algorithm is described for the solution of the underlying optimization problem, which is characterized by linear binary variables and continuous variables that appear in nonlinear functions. The proposed algorithm is based on a bounding sequence that requires the analysis of few system configurations, and the solution of a master problem that identifies new candidate structures. Application of the proposed algorithm is illustrated with the optimal synthesis of gas pipelines.

Sample average approximation for stochastic nonconvex mixed integer nonlinear programming via outer-approximation

Abstract: We propose a sample average approximation-based outer-approximation algorithm (SAAOA) that can address nonconvex two-stage stochastic programs (SP) with any continuous or discrete probability distributions. Previous work has considered this approach for convex two-stage SP (Wei and Realff in Comput Chem Eng 28(3):333–346, 2004). The SAAOA algorithm does internal sampling within a nonconvex outer-approximation algorithm where we iterate between a mixed-integer linear programming (MILP) master problem and a nonconvex nonlinear programming (NLP) subproblem. We prove that the optimal solutions and optimal value obtained by the SAAOA algorithm converge to the optimal solutions and the optimal value of the true SP problem as the sample size goes to infinity. The convergence rate is also given to estimate the sample size. Since the theoretical sample size estimate is too conservative in practice, we propose an SAAOA algorithm with confidence intervals for the upper bound and the lower bound at each iteration of the SAAOA algorithm. Two policies are proposed to update the sample sizes dynamically within the SAAOA algorithm with confidence intervals. The proposed algorithm works well for the special case of pure binary first stage variables and continuous stage two variables since in this case the nonconvex NLPs can be solved for each scenario independently. The proposed algorithm is tested with a stochastic pooling problem and is shown to outperform the external sampling approach where large scale MINLPs need to be solved.

Adaptive Robust Optimization for the Security Constrained Unit Commitment Problem

Abstract: Unit commitment, one of the most critical tasks in electric power system operations, faces new challenges as the supply and demand uncertainty increases dramatically due to the integration of variable generation resources such as wind power and price responsive demand. To meet these challenges, we propose a two-stage adaptive robust unit commitment model for the security constrained unit commitment problem in the presence of nodal net injection uncertainty. Compared to the conventional stochastic programming approach, the proposed model is more practical in that it only requires a deterministic uncertainty set, rather than a hard-to-obtain probability distribution on the uncertain data. The unit commitment solutions of the proposed model are robust against all possible realizations of the modeled uncertainty. We develop a practical solution methodology based on a combination of Benders decomposition type algorithm and the outer approximation technique. We present an extensive numerical study on the real-world large scale power system operated by the ISO New England. Computational results demonstrate the economic and operational advantages of our model over the traditional reserve adjustment approach.

An outer-approximation algorithm for a class of mixed-integer nonlinear programs

Abstract: An outer-approximation algorithm is presented for solving mixed-integer nonlinear programming problems of a particular class. Linearity of the integer (or discrete) variables, and convexity of the nonlinear functions involving continuous variables are the main features in the underlying mathematical structure. Based on principles of decomposition, outer-approximation and relaxation, the proposed algorithm effectively exploits the structure of the problems, and consists of solving an alternating finite sequence of nonlinear programming subproblems and relaxed versions of a mixed-integer linear master program. Convergence and optimality properties of the algorithm are presented, as well as a general discussion on its implementation. Numerical results are reported for several example problems to illustrate the potential of the proposed algorithm for programs in the class addressed in this paper. Finally, a theoretical comparison with generalized Benders decomposition is presented on the lower bounds predicted by the relaxed master programs.

A polyhedral branch-and-cut approach to global optimization

Abstract: A variety of nonlinear, including semidefinite, relaxations have been developed in recent years for nonconvex optimization problems. Their potential can be realized only if they can be solved with sufficient speed and reliability. Unfortunately, state-of-the-art nonlinear programming codes are significantly slower and numerically unstable compared to linear programming software.In this paper, we facilitate the reliable use of nonlinear convex relaxations in global optimization via a polyhedral branch-and-cut approach. Our algorithm exploits convexity, either identified automatically or supplied through a suitable modeling language construct, in order to generate polyhedral cutting planes and relaxations for multivariate nonconvex problems. We prove that, if the convexity of a univariate or multivariate function is apparent by decomposing it into convex subexpressions, our relaxation constructor automatically exploits this convexity in a manner that is much superior to developing polyhedral outer approximators for the original function. The convexity of functional expressions that are composed to form nonconvex expressions is also automatically exploited.Root-node relaxations are computed for 87 problems from globallib and minlplib, and detailed computational results are presented for globally solving 26 of these problems with BARON 7.2, which implements the proposed techniques. The use of cutting planes for these problems reduces root-node relaxation gaps by up to 100% and expedites the solution process, often by several orders of magnitude.