Nonconvex Generalized Benders Decomposition Research Articles

Optimization under uncertainty of a hybrid waste tire and natural gas feedstock flexible polygeneration system using a decomposition algorithm

Market uncertainties motivate the development of flexible polygeneration systems that are able to adjust operating conditions to favor production of the most profitable product portfolio. However, this operational flexibility comes at the cost of higher capital expenditure. A scenario-based two-stage stochastic nonconvex Mixed-Integer Nonlinear Programming (MINLP) approach lends itself naturally to optimizing these trade-offs. This work studies the optimal design and operation under uncertainty of a hybrid feedstock flexible polygeneration system producing electricity, methanol, dimethyl ether, olefins or liquefied (synthetic) natural gas. A recently developed C++ based software framework (named GOSSIP) is used for modeling the optimization problem as well as its efficient solution using the Nonconvex Generalized Benders Decomposition (NGBD) algorithm. Two different cases are studied: The first uses estimates of the means and variances of the uncertain parameters from historical data, whereas the second assesses the impact of increased uncertain parameter volatility. The value of implementing flexible designs characterized by the value of the stochastic solution (VSS) is in the range of 260–405 M$ for a scale of approximately 893 MW of thermal input. Increased price volatility around the same mean results in higher expected net present value and VSS as operational flexibility allows for asymmetric exploitation of price peaks.

Domain reduction for Benders decomposition based global optimization

Abstract While domain reduction has been successfully applied in branch-and-bound based global optimization over the last two decades, it has not been systematically studied for decomposition based global optimization, which is usually more efficient for problems with decomposable structures. This paper discusses integration of domain reduction in Benders decomposition based global optimization, specifically, generalized Benders decomposition (GBD) and nonconvex generalized Benders decomposition (NGBD). Revised GBD and NGBD frameworks are proposed to incorporate bound contraction operations or/and range reduction calculations, which can reduce the variable bounds and therefore improve the convergence rate and expedite the solution of nonconvex subproblems. Novel customized bound contraction problems are proposed for GBD and NGBD, and they are easier to solve than the classical bound contraction problems because they are defined on reduced variable spaces. The benefits of the proposed methods are demonstrated through a gas production operation problem and a power distribution system design problem.

Natural gas production network infrastructure development under uncertainty

Mathematical programming has been widely applied for the planning of natural gas production infrastructure development. As the production infrastructure involves large investments and is expected to remain in operation over several decades, the factors that will impact the gas production but cannot be foreseen before the development of the infrastructure need to be taken into account in the planning. Therefore, two scenario-based two-stage stochastic programming models are developed to facilitate natural gas production infrastructure development under uncertainty. One is called the stochastic pooling model, which tracks the qualities of gas streams throughout the production network via a generalized pooling model. The other is an enhancement of the stochastic pooling model with the consideration of pressure. Either model results in a large-scale nonconvex mixed-integer nonlinear programming (MINLP) problem, for which a global optimal solution, although very important for a problem that involves large investments, is very difficult to obtain. A novel optimization method, called nonconvex generalized Benders decomposition (NGBD), is developed for efficient global optimization of the large-scale nonconvex MINLP. Case studies of a real industrial natural gas production system show the advantages of the proposed stochastic programming models over deterministic optimization models, as well as the dramatic computational advantages of NGBD over a state-of-the-art global optimization solver.

Integrated crude selection and refinery optimization under uncertainty

Crude oil selection and procurement is the most important step in the refining process and impacts the profit margin of the refinery significantly. Due to uncertain quality of the crudes, conventional deterministic modeling and optimization methods are not suitable for refinery profitability enhancement. Therefore, a novel optimization scheme for crude oil procurement integrated with refinery operations in the face of uncertainties is presented. The decision process comprises two stages and is solved using a scenario‐based stochastic programming formulation. In Stage I, the optimal crude selections and purchase amounts are determined by maximizing the expected profit across all scenarios. In Stage II, the uncertainties are realized and optimal operations for the refinery are determined according to this realization. The resulting large‐scale mixed‐integer nonlinear programming formulation incorporates integer variables for crude selection and continuous variables for refinery operations, as well as bilinear terms for pooling processes. Nonconvex generalized Benders decomposition is used to solve this problem to obtain an global optimum efficiently. © 2015 American Institute of Chemical Engineers AIChE J, 62: 1038–1053, 2016

Global optimization of an industrial natural gas production network

A new stochastic programming model is proposed for integrated design and operation of natural gas production networks under uncertainty. The model not only addresses the uncertainties explicitly, but also considers the gas compositions and pressure-flow relationships for the whole production network. Due to the many nonconvex constraints, the new model leads to a very difficult large-scale nonconvex mixed-integer nonlinear programming (MINLP) problem. A modified nonconvex generalized Benders decomposition (NGBD) method is developed to efficiently solve this problem. This method integrates in NGBD several bound contraction strategies, which render tighter convex relaxations for constructing tighter lower bounding problems and accelerating the solution of nonconvex subproblems. Case study of an industrial natural gas production system shows the benefits of the proposed stochastic model and the bound contraction integrated NGBD method.

Optimal design and operation of energy systems under uncertainty

This paper is concerned with integrated design and operation of energy systems that are subject to significant uncertainties. The problem is cast as a two-stage stochastic programming problem, which can be transformed into a large-scale nonconvex mixed-integer nonlinear programming problem (MINLP). The MINLP exhibits a decomposable structure that can be exploited by nonconvex generalized Benders decomposition (NGBD) for efficient global optimization. This paper extends the NGBD method developed by the authors recently, such that the method can handle non-separable functions and integer operational decisions. Both the standard NGBD algorithm and an enhanced one with piecewise convex relaxations are discussed. The advantages of the proposed formulation and solution method are demonstrated through case studies of two industrial energy systems, a natural gas production network and a polygeneration plant. The first example shows that the two-stage stochastic programming formulation can result in better expected economic performance than the deterministic formulation, and that NGBD is more efficient than a state-of-the-art global optimization solver. The second example shows that the integration of piecewise convex relaxations can improve the efficiency of NGBD by at least an order of magnitude.

Optimal design of mixed AC–DC distribution systems for commercial buildings: A Nonconvex Generalized Benders Decomposition approach

• We develop an MINLP model which determines the optimal design of a mixed AC–DC building electrical distribution system. • We develop a tailored global optimization algorithm, based on Nonconvex Generalized Benders Decomposition. • We provide a case study, including a medium office building. Direct current (DC) electricity distribution systems have been proposed as an alternative to traditional, alternating current (AC) distribution systems for commercial buildings. Partial replacement of AC distribution with DC distribution can improve service to DC loads and overall building energy efficiency. This article develops (i) a mixed-integer, nonlinear, nonconvex mathematical programming problem to determine maximally energy efficient designs for mixed AC–DC electricity distribution systems in commercial buildings, and (ii) describes a tailored global optimization algorithm based on Nonconvex Generalized Benders Decomposition. The results of three case studies demonstrate the strength of the decomposition approach compared to state-of-the-art general-purpose global solvers.

Optimal Design and Operation of Energy Systems under Uncertainty

This paper is concerned with integrated design and operation of energy systems that are subject to significant uncertainties. The problem is cast as a two-stage stochastic nonconvex mixed-integer nonlinear program, in which the first and second stages include design decisions and operational decisions, respectively. By exploiting the separable and decomposable structure of the problem, an efficient global optimization method, called nonconvex generalized Benders decomposition (NGBD), is developed based on convex relaxation and generalized Benders decomposition. The efficiency of NGBD can be further improved via the notion of piecewise convex relaxations. The advantages of the proposed formulation and solution method are demonstrated through case studies of two industrial energy systems, a natural gas production network and a polygeneration plant. The first example shows that the stochastic programming formulation can result in better expected economic performance than the deterministic formulation, and that the NGBD solution method is dramatically more efficient than a state-of-the-art global optimization solver, especially for large numbers of scenarios. The second example further shows that the integration of piecewise convex relaxations can improve the efficiency of NGBD by at least an order of magnitude.

Parallel nonconvex generalized Benders decomposition for natural gas production network planning under uncertainty

A scenario-based two-stage stochastic programming model for gas production network planning under uncertainty is usually a large-scale nonconvex mixed-integer nonlinear programme (MINLP), which can be efficiently solved to global optimality with nonconvex generalized Benders decomposition (NGBD). This paper is concerned with the parallelization of NGBD to exploit multiple available computing resources. Three parallelization strategies are proposed, namely, naive scenario parallelization, adaptive scenario parallelization, and adaptive scenario and bounding parallelization. Case study of two industrial natural gas production network planning problems shows that, while the NGBD without parallelization is already faster than a state-of-the-art global optimization solver by an order of magnitude, the parallelization can improve the efficiency by several times on computers with multicore processors. The adaptive scenario and bounding parallelization achieves the best overall performance among the three proposed parallelization strategies.

Nonconvex Generalized Benders Decomposition with Piecewise Convex Relaxations for Global Optimization of Integrated Process Design and Operation Problems

This paper considers the global optimization of challenging stochastic or multiperiod mixed-integer nonconvex programs that arise from integrated process design and operation. The difficulties of the problems are large scale and the nonconvexity involved. Recently, a novel decomposition method, called nonconvex generalized Benders decomposition (NGBD), has been developed to solve this problem to global optimality finitely, and this method shows dramatic computational advantages over traditional branch-and-bound based global optimization methods because it can exploit well the decomposable structure of such problems. Since the convergence rate of NGBD is largely dependent on the tightness of the convex relaxations of the nonconvex functions, the efficiency of NGBD can be improved by generating tighter convex relaxations. Building on the success of piecewise linearization for bilinear programs in the process systems engineering literature, this paper develops a piecewise convex relaxation framework, which can yield tighter convex relaxations for factorable nonconvex programs, and integrates this framework into NGBD to expedite the solution. Case studies of a classical literature problem and an industry-level problem show that, while NGBD can solve problems that are intractable for a state-of-the-art global optimization solver, integrating the proposed piecewise convex relaxation into NGBD helps to reduce the solution time by up to an order of magnitude.

Nonconvex Generalized Benders Decomposition for Stochastic Separable Mixed-Integer Nonlinear Programs

This paper considers deterministic global optimization of scenario-based, two-stage stochastic mixed-integer nonlinear programs (MINLPs) in which the participating functions are nonconvex and separable in integer and continuous variables. A novel decomposition method based on generalized Benders decomposition, named nonconvex generalized Benders decomposition (NGBD), is developed to obtain ε-optimal solutions of the stochastic MINLPs of interest in finite time. The dramatic computational advantage of NGBD over state-of-the-art global optimizers is demonstrated through the computational study of several engineering problems, where a problem with almost 150,000 variables is solved by NGBD within 80 minutes of solver time.