(Global) Optimization: Historical notes and recent developments

Recent developments in (Global) Optimization are surveyed in this paper. We collected and commented quite a large number of recent references which, in our opinion, well represent the vivacity, deepness, and width of scope of current computational approaches and theoretical results about nonconvex optimization problems. Before the presentation of the recent developments, which are subdivided into two parts related to heuristic and exact approaches, respectively, we briefly sketch the origin of the discipline and observe what, from the initial attempts, survived, what was not considered at all as well as a few approaches which have been recently rediscovered, mostly in connection with machine learning

International Conference on Continuous Optimization (ICCOPT) 2019 Conference Book

The Sixth International Conference on Continuous Optimization took place on the campus of the Technical University of Berlin, August 3-8, 2019. The ICCOPT is a flagship conference of the Mathematical Optimization Society (MOS), organized every three years. ICCOPT 2019 was hosted by the Weierstrass Institute for Applied Analysis and Stochastics (WIAS) Berlin. It included a Summer School and a Conference with a series of plenary and semi-plenary talks, organized and contributed sessions, and poster sessions. This book comprises the full conference program. It contains, in particular, the scientific program in survey style as well as with all details, and information on the social program, the venue, special meetings, and more

Playing with Duality: An Overview of Recent Primal-Dual Approaches for Solving Large-Scale Optimization Problems

Optimization methods are at the core of many problems in signal/image processing, computer vision, and machine learning. For a long time, it has been recognized that looking at the dual of an optimization problem may drastically simplify its solution. Deriving efficient strategies which jointly brings into play the primal and the dual problems is however a more recent idea which has generated many important new contributions in the last years. These novel developments are grounded on recent advances in convex analysis, discrete optimization, parallel processing, and non-smooth optimization with emphasis on sparsity issues. In this paper, we aim at presenting the principles of primal-dual approaches, while giving an overview of numerical methods which have been proposed in different contexts. We show the benefits which can be drawn from primal-dual algorithms both for solving large-scale convex optimization problems and discrete ones, and we provide various application examples to illustrate their usefulness

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

Global optimization at work

In many research situations where mathematical models are used, researchers try to find parameter values such that a given performance criterion is at an optimum. If the parameters can be varied in a continuous way, this in general defines a so-called Nonlinear Programming Problem. Methods for Nonlinear Programming usually result in local optima. A local optimum is a solution (parameter values) which is the best with respect to values in the neighbourhood of that solution, not necessarily the best over the total admissible, feasible set of all possible parameter values, solutions.For mathematicians this results in the research question: How to find the best, global optimum in situations where several local optima exist?, the field of Global Optimization (GLOP). Literature, books and a specific journal, has appeared during the last decades on the field. Main focus has been on the mathematical side, i.e. given assumptions on the structure of the problems to be solved and specific global optimization methods and properties are derived. Cooperation between mathematicians and researchers (in this book called 'the modeller' or 'the potential user'), who saw global optimization problems in practical problems has lead to application of GLOP algorithms to practical optimization problems. Some of those can be found in this book. In this book we started with the question:Given a potential user with an arbitrary global optimization problem, what route can be taken in the GLOP forest to find solutions of the problem?From this first question we proceed by raising new questions. In Chapter 1 we outline the target group of users we have in mind, i.e. agricultural and environmental engineers, designers and OR workers in agricultural science. These groups are not clearly defined, nor mutually exclusive, but have in common that mathematical modelling is used and there is knowledge of linear programming and possibly of combinatorial optimization.In general, when modellers are confronted with optimization aspects, the first approach is to develop heuristics or to look for standard nonlinear programming codes to generate solutions of the optimization problem. During the search for solutions, multiple local optima may appear. We distinguished two major tracks for the path to be taken from there by the potential user to solve the problem. One track is called the deterministic track and is discussed in Chapters 2, 3 and 4. The other track is called the stochastic track and is discussed in Chapters 5 and 6. The two approaches are intended to reach a different goal.The deterministic track aims at:The global optimum is approximated (found) with certainty in a finite number of steps.The stochastic track is understood to contain some stochastic elements and aims at:Approaching the optimum in a probabilistic sense as effort grows to infinity.Both tracks are investigated in this book from the viewpoint of a potential user corresponding to the way of thinking in Popperian science. The final results are new challenging problems, questions for further research. A side question along the way is:How can the user influence the search process given the knowledge of the underlying problem and the information that becomes available during the search?The deterministic approachWhen one starts looking into the deterministic track for a given problem, one runs into the requirements which determine a major difference in applicability of the two approaches.Deterministic methods require the availability of explicit mathematical expressions of the functions to be optimized.In many practical situations which are also discussed in this book, these expressions are not available and deterministic methods cannot be applied. The operations in deterministic methods are based on concepts such as Branch-and-Bound and Cutting which require bounding of functions and parameters based on so-called mathematical structures.In Chapter 2 we describe these structures and distinguish between those which can be derived directly from the expressions, such as quadratic, bilinear and fractional functions and other structures which require analysis of the expressions such as concave and Lipschitz continuous functions. Examples are given of optimization problems revealing their structure. Moreover, we show that symmetry in the model formulation may cause models to have more than one extreme.In Chapter 3 the relationship between GLOP and Integer Programming (IP) is highlighted for several reasons.Sometimes practical GLOP problems can be approximated by IP variants and solved by standard Mixed Integer Linear Programming (MILP) techniques.The algorithms of GLOP and IP can similarly be classified.The transformability of GLOP problems to IP problems and vice versa shows that difficult problems in one class will not become easier to solve in the other.Analysis of problems, which is common in Global Optimization, can be used to better understand the complexity of some IP problems.In Chapter 4 we analyze the use of deterministic methods, demonstrating the application of the Branch-and-Bound concept. The following can be stated from the point of view of the potential user:Analysis of the expressions is required to find useful mathematical structures (Chapter 2). It should be noted that also interval arithmetic techniques can be applied directly on the expressions.The elegance of the techniques is the guarantee that we are certain about the global optimality of the optimum, when it has been discovered and verified.The methods are hard to implement. Thorough use should be made of special data structures to store the necessary information in memory.Two cases are elaborated. The quadratic product design problem illustrates how the level of Decision Support Systems can be reached for low dimensional problems, i.e. the number of variables, components or ingredients, is less than 10. The other case, the nutrient problem, shows how by analysis of the problem many useful properties can be derived which help to cut away large areas of the feasible space where the optimum cannot be situated. However, it also demonstrates the so-called Curse of Dimensionality; the problem has so many variables in a realistic situation that it is impossible to traverse the complete Branch-and-Bound tree. Therefore it is good to see the relativity of the use of deterministic methods:No global optimization method can guarantee to find and verify the global optimum for every practical situation, within a humans lifetime.The stochastic approachThe stochastic approach is followed in practice for many optimization problems by combining the generation of random points with standard nonlinear optimization algorithms. The following can be said from the point of view of the potential user.The methods require no mathematical structure of the problem and are therefore more generally applicable.The methods are relatively easy to implement.The user is never completely certain that the global optimum has been reached.The optimum is approximated in a probabilistic sense when effort increases to infinity.In Chapter 5 much attention is paid to the question what happens when a user wants to spend a limited (not infinite) amount of time to the search for the optimum, preferably less than a humans lifetime:What to do when the time for solving the problem is finite?First we looked at the information which becomes available during the search and the instruments with which the user can influence the search. It appeared that besides classical instruments which are also available in traditional nonlinear programming, the main instrument is to influence the trade-off between global (random) and local search (looking for a local optimum). This lead to a new question:Is there a best way to rule the choice between global and local search, given the information which becomes available?Analyzing in a mathematical way with extreme cases lead to the comfortable conclusion that a best method of choosing between global and local search -thus a best global optimization method- does not exist. This is valid for cases where further information (more than the information which becomes available during the search) on the function to be optimized is not available, called in literature the black-box case. The conclusion again shows that mathematical analysis with extreme cases is a powerful tool to demonstrate that so-called magic algorithms -algorithms which are said in scientific journals to be very promising, because they perform well on some test cases- can be analyzed and 'falsified' in the way of Popperian thinking. This leads to the conclusion that:Magic algorithms which are going to solve all of your problems do not exist.Several side questions derived from the main problem are investigated in this book.In Chapter 6 we place the optimization problem in the context of parameter estimation. One practical question is raised by the phenomenonEvery local search leads to a new local optimum.We know from parameter estimation that this is a symptom in so called non-identifiable systems. The minimum is obtained at a lower dimensional surface or curve. Some (non-magic) heuristics are discussed to overcome this problem.There are two side questions of users derived from the general remark:"I am not interested in the best (GLOP) solution, but in good points".The first question is that of Robust Solutions, introduced in Chapter 4, and the other is called Uniform Covering, concerning the generation of points which are nearly as good as the optimum, discussed in Chapter 6.Robust solutions are discussed in the context of product design. The robustness is defined as a measure of the error one can make from the solution so that the solution (product) is still acceptable. Looking for the most robust product is looking for that point which is as far away as possible from the boundaries of the feasible (acceptable) area. For the solution procedures, we had a look at the appearance of the problem in practice, where boundaries are given by linear and quadratic surfaces, properties of the product.For linear boundaries, finding the most robust solution is an LP problem and thus rather easy.For quadratic properties the development of specific algorithms is required.The question of Uniform Covering concerns the desire to have a set of "suboptimal" points, i.e. points with low function value (given an upper level of the function value); the points are in a so-called level set. To generate "low" points, one could run a local search many times. However, we want the points not to be concentrated in one of the compartments or one sub-area of the level set, we want them to be equally, uniformly spread over the region. This is a very difficult problem for which we test and analyze several approaches in Chapter 6. The analysis taught us that:It is unlikely that stochastic methods will be proposed which solve problems in an expected calculation time, which is polynomial in the number of variables of the problem.Final resultWhether an arbitrary problem of a user can be solved by GLOP requires analysis. There are many optimization problems which can be solved satisfactorily. Besides the selection of algorithms the user has various instruments to steer the process. For stochastic methods it mainly concerns the trade-off between local and global search. For deterministic methods it includes setting bounds and influencing the selection rule in Branch-and-Bound. We hope with this book to have given a tool and a guidance to solution procedures. Moreover, it is an introduction to further literature on the subject of Global Optimization.</p

Mixed-integer Nonlinear Optimization: a hatchery for modern mathematics

The second MFO Oberwolfach Workshop on Mixed-Integer Nonlinear Programming (MINLP) took place between 2nd and 8th June 2019. MINLP refers to one of the hardest Mathematical Programming (MP) problem classes, involving both nonlinear functions as well as continuous and integer decision variables. MP is a formal language for describing optimization problems, and is traditionally part of Operations Research (OR), which is itself at the intersection of mathematics, computer science, engineering and econometrics. The scientific program has covered the three announced areas (hierarchies of approximation, mixed-integer nonlinear optimal control, and dealing with uncertainties) with a variety of tutorials, talks, short research announcements, and a special "open problems'' session

Optimization Approaches for Electricity Generation Expansion Planning Under Uncertainty

In this dissertation, we study the long-term electricity infrastructure investment planning problems in the electrical power system. These long-term capacity expansion planning problems aim at making the most effective and efficient investment decisions on both thermal and wind power generation units. One of our research focuses are uncertainty modeling in these long-term decision-making problems in power systems, because power systems\u27 infrastructures require a large amount of investments, and need to stay in operation for a long time and accommodate many different scenarios in the future. The uncertainties we are addressing in this dissertation mainly include demands, electricity prices, investment and maintenance costs of power generation units. To address these future uncertainties in the decision-making process, this dissertation adopts two different optimization approaches: decision-dependent stochastic programming and adaptive robust optimization. In the decision-dependent stochastic programming approach, we consider the electricity prices and generation units\u27 investment and maintenance costs being endogenous uncertainties, and then design probability distribution functions of decision variables and input parameters based on well-established econometric theories, such as the discrete-choice theory and the economy-of-scale mechanism. In the adaptive robust optimization approach, we focus on finding the multistage adaptive robust solutions using affine policies while considering uncertain intervals of future demands. This dissertation mainly includes three research projects. The study of each project consists of two main parts, the formulation of its mathematical model and the development of solution algorithms for the model. This first problem concerns a large-scale investment problem on both thermal and wind power generation from an integrated angle without modeling all operational details. In this problem, we take a multistage decision-dependent stochastic programming approach while assuming uncertain electricity prices. We use a quasi-exact solution approach to solve this multistage stochastic nonlinear program. Numerical results show both computational efficient of the solutions approach and benefits of using our decision-dependent model over traditional stochastic programming models. The second problem concerns the long-term investment planning with detailed models of real-time operations. We also take a multistage decision-dependent stochastic programming approach to address endogenous uncertainties such as generation units\u27 investment and maintenance costs. However, the detailed modeling of operations makes the problem a bilevel optimization problem. We then transform it to a Mathematic Program with Equilibrium Constraints (MPEC) problem. We design an efficient algorithm based on Dantzig-Wolfe decomposition to solve this multistage stochastic MPEC problem. The last problem concerns a multistage adaptive investment planning problem while considering uncertain future demand at various locations. To solve this multi-level optimization problem, we take advantage of affine policies to transform it to a single-level optimization problem. Our numerical examples show the benefits of using this multistage adaptive robust planning model over both traditional stochastic programming and single-level robust optimization approaches. Based on numerical studies in the three projects, we conclude that our approaches provide effective and efficient modeling and computational tools for advanced power systems\u27 expansion planning