Convergent upper bounds in global minimization with nonlinear equality constraints

We address the problem of determining convergent upper bounds in continuous non-convex global minimization of box-constrained problems with equality constraints. These upper bounds are important for the termination of spatial branch-and-bound algorithms. Our method is based on the theorem of Miranda which helps to ensure the existence of feasible points in certain boxes. Then, the computation of upper bounds at the objective function over those boxes yields an upper bound for the globally minimal value. A proof of convergence is given under mild assumptions. An extension of our approach to problems including inequality constraints is possible

Branch-and-lift algorithm for deterministic global optimization in nonlinear optimal control

This paper presents a branch-and-lift algorithm for solving optimal control problems with smooth nonlinear dynamics and potentially nonconvex objective and constraint functionals to guaranteed global optimality. This algorithm features a direct sequential method and builds upon a generic, spatial branch-and-bound algorithm. A new operation, called lifting, is introduced, which refines the control parameterization via a Gram-Schmidt orthogonalization process, while simultaneously eliminating control subregions that are either infeasible or that provably cannot contain any global optima. Conditions are given under which the image of the control parameterization error in the state space contracts exponentially as the parameterization order is increased, thereby making the lifting operation efficient. A computational technique based on ellipsoidal calculus is also developed that satisfies these conditions. The practical applicability of branch-and-lift is illustrated in a numerical example. © 2013 Springer Science+Business Media New York

Disjunctive Aspects in Generalized Semi-infinite Programming

In this thesis the close relationship between generalized semi-infinite problems (GSIP) and disjunctive problems (DP) is considered. We start with the description of some optimization problems from timber industry and illustrate how GSIPs and DPs arise naturally in that field. Three different applications are reviewed. Next, theory and solution methods for both types of problems are examined. We describe a new possibility to model disjunctive optimization problems as generalized semi-infinite programs. Applying existing lower level reformulations for the obtained semi-infinite program we derive conjunctive nonlinear problems without any logical expressions, which can be locally solved by standard nonlinear solvers. In addition to this local solution procedure we propose a new branch-and-bound framework for global optimization of disjunctive programs. In contrast to the widely used reformulation as a mixed-integer program, we compute the lower bounds and evaluate the logical expression in one step. Thus, we reduce the size of the problem and work exclusively with continuous variables, which is computationally advantageous. In contrast to existing methods in disjunctive programming, none of our approaches expects any special formulation of the underlying logical expression. Where applicable, under slightly stronger assumptions, even the use of negations and implications is allowed. Our preliminary numerical results show that both procedures, the reformulation technique as well as the branch-and-bound algorithm, are reasonable methods to solve disjunctive optimization problems locally and globally, respectively. In the last part of this thesis we propose a new branch-and-bound algorithm for global minimization of box-constrained generalized semi-infinite programs. It treats the inherent disjunctive structure of these problems by tailored lower bounding procedures. Three different possibilities are examined. The first one relies on standard lower bounding procedures from conjunctive global optimization. The second and the third alternative are based on linearization techniques by which we derive linear disjunctive relaxations of the considered sub-problems. Solving these by either mixed-integer linear reformulations or, alternatively, by disjunctive linear programming techniques yields two additional possibilities. Our numerical results on standard test problems with these three lower bounding procedures show the merits of our approach

A Solver for Multiobjective Mixed-Integer Convex and Nonconvex Optimization

This paper proposes a general framework for solving multiobjective nonconvex optimization problems, i.e., optimization problems in which multiple objective functions have to be optimized simultaneously. Thereby, the nonconvexity might come from the objective or constraint functions, or from integrality conditions for some of the variables. In particular, multiobjective mixed-integer convex and nonconvex optimization problems are covered and form the motivation of our studies. The presented algorithm is based on a branch-and-bound method in the pre-image space, a technique which was already successfully applied for continuous nonconvex multiobjective optimization. However, extending this method to the mixed-integer setting is not straightforward, in particular with regard to convergence results. More precisely, new branching rules and lower bounding procedures are needed to obtain an algorithm that is practically applicable and convergent for multiobjective mixed-integer optimization problems. Corresponding results are a main contribution of this paper. What is more, for improving the performance of this new branch-and-bound method we enhance it with two types of cuts in the image space which are based on ideas from multiobjective mixed-integer convex optimization. Those combine continuous convex relaxations with adaptive cuts for the convex hull of the mixed-integer image set, derived from supporting hyperplanes to the relaxed sets. Based on the above ingredients, the paper provides a new multiobjective mixed-integer solver for convex problems with a stopping criterion purely in the image space. What is more, for the first time a solver for multiobjective mixed-integer nonconvex optimization is presented. We provide the results of numerical tests for the new algorithm. Where possible, we compare it with existing procedures

A general branch-and-bound framework for continuous global multiobjective optimization

Current generalizations of the central ideas of single-objective branch-and-bound to the multiobjective setting do not seem to follow their train of thought all the way. The present paper complements the various suggestions for generalizations of partial lower bounds and of overall upper bounds by general constructions for overall lower bounds from partial lower bounds, and by the corresponding termination criteria and node selection steps. In particular, our branch-and-bound concept employs a new enclosure of the set of nondominated points by a union of boxes. On this occasion we also suggest a new discarding test based on a linearization technique. We provide a convergence proof for our general branch-and-bound framework and illustrate the results with numerical examples

Efficient Globally Optimal Resource Allocation in Wireless Interference Networks

Radio resource allocation in communication networks is essential to achieve optimal performance and resource utilization. In modern interference networks the corresponding optimization problems are often nonconvex and their solution requires significant computational resources. Hence, practical systems usually use algorithms with no or only weak optimality guarantees for complexity reasons. Nevertheless, asserting the quality of these methods requires the knowledge of the globally optimal solution. State-of-the-art global optimization approaches mostly employ Tuy's monotonic optimization framework which has some major drawbacks, especially when dealing with fractional objectives or complicated feasible sets. In this thesis, two novel global optimization frameworks are developed. The first is based on the successive incumbent transcending (SIT) scheme to avoid numerical problems with complicated feasible sets. It inherently differentiates between convex and nonconvex variables, preserving the low computational complexity in the number of convex variables without the need for cumbersome decomposition methods. It also treats fractional objectives directly without the need of Dinkelbach's algorithm. Benchmarks show that it is several orders of magnitude faster than state-of-the-art algorithms. The second optimization framework is named mixed monotonic programming (MMP) and generalizes monotonic optimization. At its core is a novel bounding mechanism accompanied by an efficient BB implementation that helps exploit partial monotonicity without requiring a reformulation in terms of difference of increasing (DI) functions. While this often leads to better bounds and faster convergence, the main benefit is its versatility. Numerical experiments show that MMP can outperform monotonic programming by a few orders of magnitude, both in run time and memory consumption. Both frameworks are applied to maximize throughput and energy efficiency (EE) in wireless interference networks. In the first application scenario, MMP is applied to evaluate the EE gain rate splitting might provide over point-to-point codes in Gaussian interference channels. In the second scenario, the SIT based algorithm is applied to study throughput and EE for multi-way relay channels with amplify-and-forward relaying. In both cases, rate splitting gains of up to 4.5% are observed, even though some limiting assumptions have been made

Topics in exact precision mathematical programming

The focus of this dissertation is the advancement of theory and computation related to exact precision mathematical programming. Optimization software based on floating-point arithmetic can return suboptimal or incorrect resulting because of round-off errors or the use of numerical tolerances. Exact or correct results are necessary for some applications. Implementing software entirely in rational arithmetic can be prohibitively slow. A viable alternative is the use of hybrid methods that use fast numerical computation to obtain approximate results that are then verified or corrected with safe or exact computation. We study fast methods for sparse exact rational linear algebra, which arises as a bottleneck when solving linear programming problems exactly. Output sensitive methods for exact linear algebra are studied. Finally, a new method for computing valid linear programming bounds is introduced and proven effective as a subroutine for solving mixed-integer linear programming problems exactly. Extensive computational results are presented for each topic.Ph.D.Committee Chair: Dr. William J. Cook; Committee Member: Dr. George Nemhauser; Committee Member: Dr. Robin Thomas; Committee Member: Dr. Santanu Dey; Committee Member: Dr. Shabbir Ahmed; Committee Member: Dr. Zonghao G

Optimisation of temporal networks under uncertainty

A wide variety of decision problems in operations research are defined on temporal networks, that is, workflows of time-consuming tasks whose processing order is constrained by precedence relations. For example, temporal networks are used to formalise the management of projects, the execution of computer applications, the design of digital circuits and the scheduling of production processes. Optimisation problems arise in temporal networks when a decision maker wishes to determine a temporal arrangement of the tasks and/or a resource assignment that optimises some network characteristic such as the network’s makespan (i.e., the time required to complete all tasks) or its net present value. Optimisation problems in temporal networks have been investigated intensively for more than fifty years. To date, the majority of contributions focus on deterministic formulations where all problem parameters are known. This is surprising since parameters such as the task durations, the network structure, the availability of resources and the cash flows are typically unknown at the time the decision problem arises. The tacit understanding in the literature is that the decision maker replaces these uncertain parameters with their most likely or expected values to obtain a deterministic optimisation problem. It is well-documented in theory and practise that this approach can lead to severely suboptimal decisions. The objective of this thesis is to investigate solution techniques for optimisation problems in temporal networks that explicitly account for parameter uncertainty. Apart from theoretical and computational challenges, a key difficulty is that the decision maker may not be aware of the precise nature of the uncertainty. We therefore study several formulations, each of which requires different information about the probability distribution of the uncertain problem parameters. We discuss models that maximise the network’s net present value and problems that minimise the network’s makespan. Throughout the thesis, emphasis is placed on tractable techniques that scale to industrial-size problems

Two-Stage Stochastic Mixed Integer Linear Optimization

The primary focus of this dissertation is on optimization problems that involve uncertainty unfolding over time. In many real-world decisions, the decision-maker has to decide in the face of uncertainty. After the outcome of the uncertainty is observed, she can correct her initial decision by taking some corrective actions at a later time stage. These problems are known as stochastic optimization problems with recourse. In the case that the number of time stages is limited to two, these problems are referred to as two-stage stochastic optimization problems. We focus on this class of optimization problems in this dissertation. The optimization problem that is solved before the realization of uncertainty is called the first-stage problem and the problem solved to make a corrective action on the initial decision is called the second-stage problem. The decisions made in the second- stage are affected by both the first-stage decisions and the realization of random variables. Consequently, the two-stage problem can be viewed as a parametric optimization problem which involves the so-called value function of the second-stage problem. The value function describes the change in optimal objective value as the right-hand side is varied and understanding it is crucial to developing solution methods for two-stage optimization problems.In the first part of this dissertation, we study the value function of a MILP. We review the structural properties of the value function and its construction methods. We con- tribute by proposing a discrete representation of the MILP value function. We show that the structure of the MILP value function arises from two other optimization problems that are constructed from its discrete and continuous components. We show that our representation can explain certain structural properties of the MILP value function such as the sets over which the value function is convex. We then provide a simplification of the Jeroslow Formula obtained by applying our results. Finally, we describe a cutting plane algorithm for its construction and determine the conditions under which the pro- posed algorithm is finite.Traditionally, the solution methods developed for two-stage optimization problems consider the problem where the second-stage problem involves only continuous variables. In the recent years, however, two-stage problems with integer variables in the second- stage have been visited in several studies. These problems are important in practice and arise in several applications in supply chain, finance, forestry and disaster management, among others. The second part of this dissertation concerns the development and implementation of a solution method for the two-stage optimization problem where both the first and second stage involve mixed integer variables. We describe a generalization of the classical Benders’ method for solving mixed integer two-stage stochastic linear optimization problems. We employ the strong dual functions encoded in the branch-and-bound trees resulting from solution of the second-stage problem. We show that these can be used effectively within a Benders’ framework and describe a method for obtaining all required dual functions from a single, continuously refined branch-and-bound tree that is used to warm start the solution procedure for each subproblem.Finally, we provide details on the implementation of our proposed algorithm. The implementation allows for construction of several approximations of the value function of the second-stage problem. We use different warm-starting strategies within our proposed algorithm to solve the second-stage problems, including solving all second-stage problems with a single tree. We provide computational results on applying these strategies to the stochastic server problems (SSLP) from the stochastic integer programming test problem library (SIPLIB)