Deterministic Global Optimization Approach Research Articles

A rigorous deterministic global optimization approach for the derivation of secondary information in digital maps

We derive a generic system that constructs an optimization model for an emergency stop scenario on the highway, based on map data from high definition maps that are used in Advanced Driver Assistance Systems (ADAS) and in Highly Automated Driving (HAD). New additional situative and scenario-based information is computed by applying a global maximization approach to the model. For this purpose, we develop two new rigorous and deterministic branch-and-bound algorithms that both determine the certified global optimal value up to a predefined tolerance. The underlying interval optimization algorithm, which uses first-order techniques, is enhanced by one of two second-order methods that are applied for specifically selected intervals. We investigate two approaches that either compute a concave overestimator for the objective function or approximate the function with a quadratic polynomial using Taylor expansion. We show the limits of interval arithmetic in our problem, especially for the interval versions of the derivatives, and present a local linearization of the curve data that improves the results significantly. The presented novel method for deriving secondary information is compared to state of the art methods on two exemplary and for the automotive context representative scenarios to show the advantages of our approach.

Operational strategy and planning for raw natural gas refining complexes: Process modeling and global optimization

Optimal operational strategy and planning of a raw natural gas refining complex (RNGRC) is very challenging since it involves highly nonlinear processes, complex thermodynamics, blending, and utility systems. In this article, we first propose a superstructure integrating a utility system for the RNGRC, involving multiple gas feedstocks, and different product specifications. Then, we develop a large‐scale nonconvex mixed‐integer nonlinear programming (MINLP) optimization model. The model incorporates rigorous process models for input and output relations based on fundamentals of thermodynamics and unit operations and accurate models for utility systems. To reduce the noncovex items in the proposed MINLP model, equivalent reformulation techniques are introduced. Finally, the reformulated nonconvex MINLP model is solved to global optimality using state of the art deterministic global optimization approaches. The computational results demonstrate that a significant profit increase is achieved using the proposed approach compared to that from the real operation. © 2016 American Institute of Chemical Engineers AIChE J, 63: 652–668, 2017

Integrated gasoline blending and order delivery operations: Part I. short‐term scheduling and global optimization for single and multi‐period operations

Gasoline is one of the most valuable products in an oil refinery and can account for as much as 60–70% of total profit. Optimal integrated scheduling of gasoline blending and order delivery operations can significantly increase profit by avoiding ship demurrage, improving customer satisfaction, minimizing quality give‐aways, reducing costly transitions and slop generation, exploiting low‐quality cuts, and reducing inventory costs. In this article, we first introduce a new unit‐specific event‐based continuous‐time formulation for the integrated treatment of recipes, blending, and scheduling of gasoline blending and order delivery operations. Many operational features are included such as nonidentical parallel blenders, constant blending rate, minimum blend length and amount, blender transition times, multipurpose product tanks, changeovers, and piecewise constant profiles for blend component qualities and feed rates. To address the nonconvexities arising from forcing constant blending rates during a run, we propose a hybrid global optimization approach incorporating a schedule adjustment procedure, iteratively via a mixed‐integer programming and nonlinear programming scheme, and a rigorous deterministic global optimization approach. The computational results demonstrate that our proposed formulation does improve the mixed‐integer linear programming relaxation of Li and Karimi, Ind. Eng. Chem. Res., 2011, 50, 9156–9174. All examples are solved to be 1%‐global optimality with modest computational effort. © 2016 American Institute of Chemical Engineers AIChE J, 62: 2043–2070, 2016

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.

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 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).

An improved hybrid global optimization method for protein tertiary structure prediction

First principles approaches to the protein structure prediction problem must search through an enormous conformational space to identify low-energy, near-native structures. In this paper, we describe the formulation of the tertiary structure prediction problem as a nonlinear constrained minimization problem, where the goal is to minimize the energy of a protein conformation subject to constraints on torsion angles and interatomic distances. The core of the proposed algorithm is a hybrid global optimization method that combines the benefits of the αBB deterministic global optimization approach with conformational space annealing. These global optimization techniques employ a local minimization strategy that combines torsion angle dynamics and rotamer optimization to identify and improve the selection of initial conformations and then applies a sequential quadratic programming approach to further minimize the energy of the protein conformations subject to constraints. The proposed algorithm demonstrates the ability to identify both lower energy protein structures, as well as larger ensembles of low-energy conformations.

Deterministic global optimization approach to steady-state distribution gas pipeline networks

Natural gas is normally transported through a vast network of pipelines. A pipeline network is generally established either to transmit gas at high pressure from coastal supplies to regional demand points (transmission network) or to distribute gas to consumers at low pressure from the regional demand points (distribution network). In this study, the distribution network is considered. The distribution network differs from the transmission one in a number of ways. Pipes involved in a distribution network are often much smaller and the network is simpler, having no valves, compressors or nozzles. In this paper, we propose the problem of minimizing the cost of pipelines incurred by driving the gas in a distribute non-linear network under steady-state assumptions. In particular, the decision variables include the length of the pipes’ diameter, pressure drops at each node of the network, and mass flow rate at each pipeline leg. We establish a mathematical optimization model of this problem, and then present a global approach, which is based on the GOP primal-relaxed dual decomposition method presented by Visweswaran and Floudas (Global optimization in engineering design. Kluwer book series in nonconvex optimization and its applications. Kluwer, Netherlands, 1996), to the optimization model. Finally, results from application of the approach to data from gas company are presented.

Real-Coded Genetic Algorithm for Solving Generalized Polynomial Programming Problems

Generalized polynomial programming (GPP) is a nonlinear programming (NLP) method based on a nonconvex objective function, which is subject to nonconvex inequality constraints. Hence, a GPP problem has multiple local optima in its constrained solution space. General NLP techniques use local optimization, and therefore do not easily solve GPP problems. Some deterministic global optimization approaches have been developed to overcome this drawback of NLP methods. Although these approaches yield a global solution to a GPP problem, they can be mathematically tedious. Therefore, this study presents a real-coded genetic algorithm (RGA), which is a stochastic global optimization method, to find a global solution to a GPP problem. The proposed RGA is used to solve a set of GPP problems. The best solution obtained by the RGA is compared with the known global solution to each test problem. Numerical results show that the proposed RGA converges to a global solution to a GPP problem.

Packing equal circles in a square: a deterministic global optimization approach

In this paper the problem of packing n equal circles into the unit square will be considered. Starting from a general rectangular branch-and-bound algorithm, many tools, which exploit the special structure of the problem and properties fulfilled by some of its solutions, will be introduced and discussed. Computational results will be presented and, in particular, the optimality within a given tolerance of best known solutions in the literature for n=10–35, n=38,39 will be proved, with the exception of the case n=32 for which a new solution has been detected and proved to be optimal within the given tolerance. Moreover, a new solution for n=37 has been detected, but not yet proved to be optimal within the given tolerance.

Locating All Heterogeneous and Reactive Azeotropes in Multicomponent Mixtures

A novel approach for enclosing all heterogeneous and reactive azeotropes in multicomponent mixtures is presented. The thermodynamic conditions for azeotropy form a system of nonlinear equations. A deterministic global optimization approach is introduced in which the global optimization problem may contain multiple global minima and there is a one-to-one correspondence between each global minimum and a potential azeotrope. An algorithm is presented which provides a theoretical guarantee that all global minima can be enclosed. This method is general and can be applied to multicomponent mixtures that are described by standard thermodynamic models. The problem formulations for finding all heterogeneous azeotropes and for finding all reactive azeotropes are presented. The effectiveness of the proposed approach is illustrated through several example problems using the Wilson, NRTL, UNIQUAC, and UNIFAC activity coefficient equations.

Global optimization in design and control of chemical process systems

Abstract This paper presents an overview of the recent advances in deterministic global optimization approaches and their applications in the areas of Process Design and Control. The focus is on global optimization methods for (a) twice-differentiable constrained nonlinear optimization problems, (b) mixed-integer nonlinear optimization problems, and (c) locating all solutions of nonlinear systems of equations. Theoretical advances and computational studies on process design, batch design under uncertainty, phase equilibrium, location of azeotropes, stability margin, process synthesis, and parameter estimation problems are discussed.

Deterministic Global Optimization in Nonlinear Optimal Control Problems

The accurate solution of optimal control problems is crucial in many areas of engineering and applied science. For systems which are described by a nonlinear set of differential-algebraic equations, these problems have been shown to often contain multiple local minima. Methods exist which attempt to determine the global solution of these formulations. These algorithms are stochastic in nature and can still get trapped in local minima. There is currently no deterministic method which can solve, to global optimality, the nonlinear optimal control problem. In this paper a deterministic global optimization approach based on a branch and bound framework is introduced to address the nonlinear optimal control problem to global optimality. Only mild conditions on the differentiability of the dynamic system are required. The implementa-tion of the approach is discussed and computational studies are presented for four control problems which exhibit multiple local minima.

Global Optimization in Design and Control of Chemical Process Systems

This paper presents an overview of the recent advances in deterministic global optimization approaches and their applications in the areas of Process Design and Control. The focus is on global optimization methods for (a) twice-differentiable constrained nonlinear optimization problems, (b) mixed-integer nonlinear optimization problems, and (c) locating all solutions of nonlinear systems of equations. Theoretical advances and computational studies on process design, batch design under uncertainty, phase equilibrium, location of azeotropes, stability margin, process synthesis, and parameter estimation problems are discussed.

A deterministic global optimization approach for molecular structure determination

A deterministic global optimization algorithm is introduced for locating global minimum potential energy molecular conformations. The proposed branch and bound type algorithm attains finite ε-convergence to the global minimum through the successive refinement of converging lower and upper bounds on the solution. These bounds are obtained through a novel convex lowering bounding of the total potential function and the subsequent solution of a series of nonlinear convex optimization problems. The minimization of the total potential energy function is performed on an independent set of internal coordinates involving only dihedral angles. A number of example problems illustrate the proposed approach.

Primal-relaxed dual global optimization approach

A deterministic global optimization approach is proposed for nonconvex constrained nonlinear programming problems. Partitioning of the variables, along with the introduction of transformation variables, if necessary, converts the original problem into primal and relaxed dual subproblems that provide valid upper and lower bounds respectively on the global optimum. Theoretical properties are presented which allow for a rigorous solution of the relaxed dual problem. Proofs of ∈-finite convergence and ∈-global optimality are provided. The approach is shown to be particularly suited to (a) quadratic programming problems, (b) quadratically constrained problems, and (c) unconstrained and constrained optimization of polynomial and rational polynomial functions. The theoretical approach is illustrated through a few example problems. Finally, some further developments in the approach are briefly discussed.

New properties and computational improvement of the GOP algorithm for problems with quadratic objective functions and constraints

In Floudas and Visweswaran (1990, 1993), a deterministic global optimization approach was proposed for solving certain classes of nonconvex optimization problems. An algorithm, GOP, was presented for the solution of the problem through a series ofprimal andrelaxed dual problems that provide valid upper and lower bounds respectively on the global solution. The algorithm was proved to have finite convergence to an ∈-global optimum. In this paper, new theoretical properties are presented that help to enhance the computational performance of the GOP algorithm applied to problems of special structure. The effect of the new properties is illustrated through application of the GOP algorithm to a difficult indefinite quadratic problem, a multiperiod tankage quality problem that occurs frequently in the modeling of refinery processes, and a set of pooling/blending problems from the literature. In addition, extensive computational experience is reported for randomly generated concave and indefinite quadratic programming problems of different sizes. The results show that the properties help to make the algorithm computationally efficient for fairly large problems.

A global optimization algorithm (GOP) for certain classes of nonconvex NLPs—II. Application of theory and test problems

In Part I (Floudas and Visweswaran, Computers chem. Engng 14, 1397, 1990), a deterministic global optimization approach was proposed for solving certain classes of nonconvex optimization problems. An algorithm, GOP, was presented for the rigorous solution of the problem through a series of primal and relaxed dual problems until the upper and lower bounds from these problems converged to an ε -global optimum. In this paper, theoretical results are presented for several classes of mathematical programming problems that include: (i) the general quadratic programming problem; (ii) quadratic programming problems with quadratic constraints; (iii) pooling and blending problems; and (iv) unconstrained and constrained optimization problems with polynomial terms in the objective function and/or constraints. For each class, a few examples are presented illustrating the approach.