Deterministic Global Optimization Research Articles

A generalization of the Branch-and-Sandwich algorithm: From continuous to mixed-integer nonlinear bilevel problems

We propose a deterministic global optimization algorithm for mixed-integer nonlinear bilevel problems (MINBP) by generalizing the Branch-and-Sandwich algorithm (Kleniati and Adjiman, 2014a). Advances include the removal of regularity assumptions and the extension of the algorithm to mixed-integer problems. The proposed algorithm can solve very general MINBP problems to global optimality, including problems with inner equality constraints that depend on the inner and outer variables. Inner lower and inner upper bounding problems are constructed to bound the inner optimal value function and provide constant-bound cuts for the outer bounding problems. To remove the need for regularity, we introduce a robust counterpart approach for the inner upper bounding problem. Branching is allowed on all variables without distinction by keeping track of refined partitions of the inner space for every refined subdomain of the outer space. Finite ɛ-convergence to the global solution is proved. The algorithm is applied successfully to 10 mixed-integer literature problems.

Coal and Biomass to Liquid Transportation Fuels: Process Synthesis and Global Optimization Strategies

The thermochemical conversion of coal and biomass to liquid transportation fuels from a synthesis gas intermediate is investigated using an optimization-based process synthesis framework. Two distinct types of coal (LV bituminous and coal commonly found in the province of Anhui, China) and two types of biomass (hardwood and duckweed) are considered as feedstocks. The superstructure incorporates alternative conversion pathways of synthesis gas which include methanol formation and conversion into FischerTropsch hydrocarbons. Methanol may be converted to gasoline or olefins, and the olefins may be subsequently converted to gasoline and distillate. A rigorous deterministic global optimization branch-and-bound framework is utilized to determine the optimal process topology that produces liquid fuels at the lowest possible cost. Economies of scale are evident as the refinery capacity increases and it is observed that the fuel ratios of the final liquid products have a significant impact on the optimal topology of the plant. The results suggest that liquid fuels production from coal and biomass can be competitive with petroleum-based processes.

Outer Approximation and Outer-Inner Approximation Approaches for Unit Commitment Problem

This paper proposes a new separable model for the unit commitment (UC) problem and three deterministic global optimization methods for it ensuring convergence to the global optimum within a desired tolerance. By decomposing a multivariate function into several univariate functions, a tighter outer approximation methodology that can be used to improve the outer approximations of several classical convex programming techniques is presented. Based on the idea of the outer approximation (OA) method and the proposed separable model, an outer-inner approximation (OIA) approach is also presented to solve this new formulation of UC problem. In this OIA approach, the UC problem is decomposed into a tighter outer approximation subproblem and an inner approximation subproblem, where the former leads to a better lower bound than the OA method, and the later provides a better upper bound. The simulation results for systems of up to 100 units with 24 h are compared with those of previously published methods, which show that the OIA approach is very promising due to the excellent performance. The proposed approaches are also applied to the large-scale systems of up to 1000 units with 24 h, and systems of up to 100 units with 96 h and 168 h.

A new linearization technique for minimax linear fractional programming

This paper presents a deterministic global optimization algorithm for solving minimax linear fractional programming (MLFP). In this algorithm, a new linearization technique is proposed, which uses more information of the objective function of the (MLFP) than other techniques. By utilizing this new linearization technique, the initial nonconvex programming problem (MLFP) is reduced to a sequence of linear relaxation programming (LRP) problems, which can provide reliable lower bound of the optimal value. The proposed algorithm is convergent to the global minimum of the (MLFP) through the successive refinement of the feasible region and solutions of a series of the (LRP). Compared with the known algorithms, numerical results show that the proposed algorithm is robust and effective.

Branch-and-Sandwich: a deterministic global optimization algorithm for optimistic bilevel programming problems. Part I: Theoretical development

We present a global optimization algorithm, Branch-and-Sandwich, for optimistic bilevel programming problems that satisfy a regularity condition in the inner problem. The functions involved are assumed to be nonconvex and twice continuously differentiable. The proposed approach can be interpreted as the exploration of two solution spaces (corresponding to the inner and the outer problems) using a single branch-and-bound tree. A novel branching scheme is developed such that classical branch-and-bound is applied to both spaces without violating the hierarchy in the decisions and the requirement for (global) optimality in the inner problem. To achieve this, the well-known features of branch-and-bound algorithms are customized appropriately. For instance, two pairs of lower and upper bounds are computed: one for the outer optimal objective value and the other for the inner value function. The proposed bounding problems do not grow in size during the algorithm and are obtained from the corresponding problems at the parent node.

Branch-and-Sandwich: a deterministic global optimization algorithm for optimistic bilevel programming problems. Part II: Convergence analysis and numerical results

In the first part of this work, we presented a global optimization algorithm, Branch-and-Sandwich, for optimistic bilevel programming problems that satisfy a regularity condition in the inner problem (Kleniati and Adjiman in J Glob Optim, 2014). The proposed approach can be interpreted as the exploration of two solution spaces (corresponding to the inner and the outer problems) using a single branch-and-bound tree, where two pairs of lower and upper bounds are computed: one for the outer optimal objective value and the other for the inner value function. In the present paper, the theoretical properties of the proposed algorithm are investigated and finite $$\varepsilon $$ -convergence to a global solution of the bilevel problem is proved. Thirty-four problems from the literature are tackled successfully.

On the Computational Studies of Deterministic Global Optimization of Head Dependent Short-Term Hydro Scheduling

This paper addresses the global optimization of the short term scheduling for hydroelectric power generation. A tailored deterministic global optimization approach, denominated sHBB, is developed and its performance is analyzed. This approach is applied to the optimization of a mixed integer nonlinear programming (MINLP) model for cascades of hydro plants, each one with multiple turbines, and characterized by a detailed representation of the net head of water, and a nonlinear hydropower generation function. A simplified model is also considered where only the linear coefficients of the forebay and tailrace polynomial functions are retained. For comparison purposes, four case studies are addressed with the proposed global optimization strategy and with a commercial solver for global optimization. The results show that the proposed approach is more efficient than the commercial solver in terms of finding a better solution with a smaller optimality gap, using less CPU time. The proposed method can also find alternative and potentially more profitable power production schedules. Significant insights were also obtained regarding the effectiveness of the proposed relaxation strategies.

Joint Optimization of Source Power Allocation and Cooperative Beamforming for SC-FDMA Multi-User Multi-Relay Networks

This paper is concerned with design problems of joint source power allocation and relay beamforming in multi-user multi-relay networks that use single-carrier frequency division multiple access (SC-FDMA) and amplify-and-forward relaying. Examined are the joint programs of (i) maximizing the minimum signal-to-interference-plus-noise ratio (SINR) under various transmitted power constraints, and (ii) minimizing the total transmitted power subject to prescribed SINR thresholds of users. Although these optimization problems are highly nonconvex and have large dimensions, by exploiting their partial convexities and making elegant nonlinear variable changes, they are recast as d.c. (difference of two convex) programs. Efficient d.c. iterative procedures are then developed to find the solutions. Simplified joint programs under the two cases of equal source power and equal relay beamforming weights, respectively, are also considered. Branch-and-bound algorithms of deterministic global optimization are then proposed for solving the simplified joint programs. Simulation results confirm the excellent performance and computational efficiency of all the proposed solutions.

Assessment of a non-adaptive deterministic global optimization algorithm for problems with low-dimensional non-convex subspaces

The optimum and at least one optimizing point for convex nonlinear programs can be approximated well by the solution to a linear program (a fact long used in branch and bound algorithms). In more general problems, we can identify subspaces of ‘non-convex variables’ such that, if these variables have sufficiently small ranges, the optimum and at least one optimizing point can be approximated well by the solution of a single linear program. If these subspaces are low-dimensional, this suggests subdividing the variables in the subspace a priori, then producing and solving a fixed, known number of linear programs to obtain an approximation to the solution. The total amount of computation is much more predictable than that required to complete a branch and bound algorithm, and the scheme is ‘embarrassingly parallel’, with little need for either communication or load balancing. We compare such a non-adaptive scheme experimentally to our GlobSol branch and bound implementation, on those problems from the COCONUT project Lib1 test set with non-convex subspaces of dimension four or less, and we discuss potential alterations to both the non-adaptive scheme and our branch and bound process that might change the scope of applicability.

Global optimization of bounded factorable functions with discontinuities

A deterministic global optimization method is developed for a class of discontinuous functions. McCormick’s method to obtain relaxations of nonconvex functions is extended to discontinuous factorable functions by representing a discontinuity with a step function. The properties of the relaxations are analyzed in detail; in particular, convergence of the relaxations to the function is established given some assumptions on the bounds derived from interval arithmetic. The obtained convex relaxations are used in a branch-and-bound scheme to formulate lower bounding problems. Furthermore, convergence of the branch-and-bound algorithm for discontinuous functions is analyzed and assumptions are derived to guarantee convergence. A key advantage of the proposed method over reformulating the discontinuous problem as a MINLP or MPEC is avoiding the increase in problem size that slows global optimization. Several numerical examples for the global optimization of functions with discontinuities are presented, including ones taken from process design and equipment sizing as well as discrete-time hybrid systems.

Biomass and Natural Gas to Liquid Transportation Fuels: Process Synthesis, Global Optimization, and Topology Analysis

An optimization-based process synthesis framework is proposed for the thermochemical conversion of biomass and natural gas to liquid fuels (BGTL). Hydrocarbons are produced from synthesis gas either directly via Fischer–Tropsch synthesis or indirectly via catalytic conversion of methanol over ZSM-5. Different conversion technologies are investigated to examine their economic effects on BGTL refineries that produce gasoline, diesel, and kerosene. The process synthesis framework includes simultaneous heat, power, and water integration and utilizes a rigorous deterministic global optimization strategy to mathematically guarantee the minimal cost of the BGTL refineries. The refineries have at least 50% less life-cycle CO2 emissions than a standard petroleum-based refinery. Forty-eight case studies are presented to determine the effect of refinery capacity, liquid fuel composition, and biomass feedstock on the overall cost, topological design, material/energy balances, and life-cycle greenhouse gas emissions. Results suggest that these systems can be economically competitive with petroleum-based processes while achieving the 50% emissions reduction.

Book Reviews

In Book Reviews, we review an extensive and diverse range of books. They cover theory and applications in operations research, statistics, management science, econometrics, mathematics, computers, and information systems. In addition, we include books in other fields that emphasize technical applications. The editor will be pleased to receive an e-mail from those willing to review a book, with an indication of specific areas of interest. If you are aware of a specific book that you would like to review, or that you think should be reviewed, please contact the editor. The following books are reviewed in this issue of Interfaces, 43(1), January–February 2013: Supply Chain Simulation: A System Dynamics Approach for Improving Performance, Francisco Campuzano and Josefa Mula; Models Behaving Badly, Emanuel Derman; Design of Modern Heuristics, Franz Rothlauf; Deterministic Global Optimization: Geometric Branch-and-Bound Methods and Their Applications, Daniel Scholz; Project Management with Dynamic Scheduling—Baseline Scheduling, Risk Analysis and Project Control, Mario Vanhoucke; Optimal Control Theory with Applications in Economics, Thomas A. Weber.

Novel Natural Gas to Liquids Processes: Process Synthesis and Global Optimization Strategies

An optimization‐based process synthesis framework is proposed for the conversion of natural gas to liquid transportation fuels. Natural gas conversion technologies including steam reforming, autothermal reforming, partial oxidation to methanol, and oxidative coupling to olefins are compared to determine the most economic processing pathway. Hydrocarbons are produced from Fischer–Tropsch (FT) conversion of syngas, ZSM‐5 catalytic conversion of methanol, or direct natural gas conversion. Multiple FT units with different temperatures, catalyst types, and hydrocarbon effluent compositions are investigated. Gasoline, diesel, and kerosene are generated through upgrading units involving carbon‐number fractionation or ZSM‐5 catalytic conversion. A powerful deterministic global optimization method is introduced to solve the mixed‐integer nonlinear optimization model that includes simultaneous heat, power, and water integration. Twenty‐four case studies are analyzed to determine the effect of refinery capacity, liquid fuel composition, and natural gas conversion technology on the overall system cost, the process material/energy balances, and the life cycle greenhouse gas emissions. © 2013 American Institute of Chemical Engineers AIChE J, 59: 505–531, 2013

Convex and Concave Relaxations for the Parametric Solutions of Semi-explicit Index-One Differential-Algebraic Equations

A method is presented for computing convex and concave relaxations of the parametric solutions of nonlinear, semi-explicit, index-one differential-algebraic equations (DAEs). These relaxations are central to the development of a deterministic global optimization algorithm for problems with DAEs embedded. The proposed method uses relaxations of the DAE equations to derive an auxiliary system of DAEs, the solutions of which are proven to provide the desired relaxations. The entire procedure is fully automatable.

Geometric branch-and-bound methods for constrained global optimization problems

Geometric branch-and-bound methods are popular solution algorithms in deterministic global optimization to solve problems in small dimensions. The aim of this paper is to formulate a geometric branch-and-bound method for constrained global optimization problems which allows the use of arbitrary bounding operations. In particular, our main goal is to prove the convergence of the suggested method using the concept of the rate of convergence in geometric branch-and-bound methods as introduced in some recent publications. Furthermore, some efficient further discarding tests using necessary conditions for optimality are derived and illustrated numerically on an obnoxious facility location problem.

Deterministic and stochastic global optimization techniques for planar covering with ellipses problems

Problems of planar covering with ellipses are tackled in this work. Ellipses can have a fixed angle or each of them can be freely rotated. Deterministic global optimization methods are developed for both cases, while a stochastic version of the method is also proposed for large instances of the latter case. Numerical results show the effectiveness and efficiency of the proposed methods.

A general framework for convexity analysis in deterministic global optimization

In previous work, we, and also Epperly and Pistikopoulos, proposed an analysis of general nonlinear programs that identified certain variables as convex, not ever needing subdivision, and non-convex, or possibly needing subdivision in branch and bound algorithms. We proposed a specific algorithm, based on a generated computational graph of the problem, for identifying such variables. In our previous work, we identified only independent variables in the computational graph. Here, we examine alternative sets of non-convex variables consisting not just of independent variables, but of a possibly smaller number of intermediate variables. We do so with examples and theorems. We also apply variants of our proposed analysis to the well-known COCONUT Lib-1 test set. If the number of such non-convex variables is sufficiently small, it may be possible to fully subdivide them before analysis of ranges of objective and constraints, thus dispensing totally with the branch and bound process. Advantages to such a non-adaptive process include higher predictability and easier parallizability. We present an algorithm and exploratory results here, with a more complete empirical study in a subsequent paper.

Gene regulatory network modeling via global optimization of high-order dynamic Bayesian network

BackgroundDynamic Bayesian network (DBN) is among the mainstream approaches for modeling various biological networks, including the gene regulatory network (GRN). Most current methods for learning DBN employ either local search such as hill-climbing, or a meta stochastic global optimization framework such as genetic algorithm or simulated annealing, which are only able to locate sub-optimal solutions. Further, current DBN applications have essentially been limited to small sized networks.ResultsTo overcome the above difficulties, we introduce here a deterministic global optimization based DBN approach for reverse engineering genetic networks from time course gene expression data. For such DBN models that consist only of inter time slice arcs, we show that there exists a polynomial time algorithm for learning the globally optimal network structure. The proposed approach, named GlobalMIT+, employs the recently proposed information theoretic scoring metric named mutual information test (MIT). GlobalMIT+ is able to learn high-order time delayed genetic interactions, which are common to most biological systems. Evaluation of the approach using both synthetic and real data sets, including a 733 cyanobacterial gene expression data set, shows significantly improved performance over other techniques.ConclusionsOur studies demonstrate that deterministic global optimization approaches can infer large scale genetic networks.

Global optimization of actively morphing flapping wings

We consider active shape morphing to optimize the flight performance of flapping wings. To this end, we combine a three-dimensional version of the unsteady vortex lattice method (UVLM) with a deterministic global optimization algorithm to identify the optimal kinematics that maximize the propulsive efficiency under lift and thrust constraints. The UVLM applies only to incompressible, inviscid flows where the separation lines are known a priori. Two types of morphing parameterization are investigated here—trigonometric and spline-based. The results show that the spline-based morphing, which requires specification of more design variables, yields a significant improvement in terms of propulsive efficiency. Furthermore, we remark that the average value of the lift coefficient in the optimized kinematics remained equal to the value in the baseline case (without morphing). This indicates that morphing is most efficiently used to generate thrust and not to increase lift beyond the basic value obtained by flapping only. Besides, our study gives comparable optimal efficiencies to those obtained from previous studies based on gradient-based optimization, but completely different design points (especially for the spline-based morphing), which would indicate that the design space associated with the flapping kinematics is very complex.

Global optimization of mixed-integer quadratically-constrained quadratic programs (MIQCQP) through piecewise-linear and edge-concave relaxations

We propose a deterministic global optimization approach, whose novel contributions are rooted in the edge-concave and piecewise-linear underestimators, to address nonconvex mixed-integer quadratically-constrained quadratic programs (MIQCQP) to \({\epsilon}\) -global optimality. The facets of low-dimensional (n ≤ 3) edge-concave aggregations dominating the termwise relaxation of MIQCQP are introduced at every node of a branch-and-bound tree. Concave multivariable terms and sparsely distributed bilinear terms that do not participate in connected edge-concave aggregations are addressed through piecewise-linear relaxations. Extensive computational studies are presented for point packing problems, standard and generalized pooling problems, and examples from GLOBALLib (Meeraus, Globallib. http://www.gamsworld.org/global/globallib.htm).