A decomposition method for large scale MILPs, with performance guarantees and a power system application☆
Introduction
In this paper we investigate mixed-integer optimization problems in the form We refer to as the resource vector, and to the sets as the subsystems. We assume that each of the sets is a non-empty, compact, mixed-integer polyhedral set that can be written as with and . We further assume that the problem is feasible and that the total number of subsystems is greater than the length of the resource vector. Our principal interest is in large-scale optimization problems, i.e. those for which , while remaining finite.
Problem can be viewed generically as modeling any problem for which a large number of subproblems defined on the domains , whose description can include integer variables, are coupled through a small number of complicating constraints . These coupling constraints determine the limits on the available resources to be shared among the subsystems. Simple examples of problems in this form include classical combinatorial programs such as the multidimensional knapsack problem, in which , and (Wilbaut, Hanafi, & Salhi, 2008).
More complicated instances of problems in the form , with more detailed models for the subsystems , arise in a variety of contexts. In power systems, scheduling the operation of power generation plants (Yamin, 2004) is a decision problem in which the subsystems are the generating units, integer variables in the local models arise due to, e.g., start-up and shut-down costs, and the coupling constraints are related to the requirement that generation must match load. In supply chain management, models fitting appear in the problem of partial shipments (Dawande et al., 2006, Vujanic et al., in press-b). Portfolio optimization for small investors, for which mixed-integer models have been proposed, is another example application (Baumann & Trautmann, 2013). Finally, some sparse problems that do not naturally possess the structure of can be reformulated to fit our framework by appropriately permuting rows and columns of the constraints matrix; Bergner et al. (2011) propose a method to automate this procedure.
A direct solution of is typically problematic when the problem is very large, since the problem amounts to a mixed-integer linear program of possibly very large size. As a result, the Lagrange dual of is often taken as a useful alternative, because the resulting dual problem is separable in the subsystems despite the presence of the complicating constraints. When this dual problem is solved by an iterative method, e.g. using the subgradient method (Bertsekas, 1999), a candidate (primal) solution to can be computed at each iteration.
For problems affected by non-zero duality gap such as , however, this approach suffers from a major drawback. Namely, any guarantee about the properties of these candidate solutions is lost. Even at the dual optimal solution, the associated candidate primal solutions may be suboptimal and can even be infeasible.
The principal goal of this paper is to propose a new solution method for problem that preserves the attractive features of solution via the Lagrange dual, while at the same time protecting the recovered primal solutions from infeasibility.
Literature. Lagrangian relaxation for mixed integer programs was first introduced by Held and Karp (1970), and many of its theoretical properties were described in Geoffrion (1974). Properties of the inner solutions in the convex case are well known (Rockafellar, 1997, Thm. 28.1). It is also well known that in general these properties are lost in the mixed-integer case (Bertsekas, 1999, Section 5.5.3). Because of this, primal recovery methods based on Lagrangian duality are often two-phase schemes in which an infeasible solution is found through duality in the first stage, and in the second stage it is rectified into a feasible one using heuristics, see, e.g., Bertsekas, Lauer, Sandell, and Posbergh (1983) and Redondo and Conejo (1999).
Duality for problems specifically in the form has been studied at least as early as in Aubin and Ekeland (1976), where some of its special features were first characterized. In particular, it was noted that the duality gap for this program structure decreases in relative terms as the problem increases in size, as measured by the cardinality of . We will show that the mechanism behind this vanishing gap effect can also be used to recover “good” primal solutions for the mixed-integer program directly from the dual, in a way that resembles the convex (zero gap) case.
In practical applications, this behavior of the duality gap has been observed in Bertsekas et al. (1983) in the context of unit commitments for power systems. In this case it is exploited in an algorithm that provides solutions to the extended master problem, but no connection to the solutions of the inner problem is provided. It also appears in the multistage stochastic integer programming literature (Birge and Dempstert, 1996, Caroe and Schultz, 1999), where it is used to gauge the strength of the Lagrangian relaxation, but in which no relations to primal solutions are drawn. Another domain in which diminishing gap has been used is in communications, more precisely in optimization of multicarrier communication systems (Yu & Lui, 2006). However, in this case non-convexity is in the objective function rather than due to the presence of integer variables.
Current contribution. In this paper we further investigate duality for programs structured as and focus on the primal solutions recovered at the dual optimum.
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We provide a new relation between the optimizers of a convexified form of and the solutions obtained from the dual problem. This relation holds under mild conditions that are commonly satisfied in practice.
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Using this relation we can bound the magnitude of the constraint violations of the solutions recovered from the dual. In light of this bound, we propose a new solution method which is guaranteed to produce feasible solutions for . The method is based on an appropriate contraction of the resources .
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We also provide a performance bound of the solutions recovered, which indicates that their quality improves as the problem size increases. For particular structures, arising e.g. from underlying physical networks, we refine our theoretical results to improve the performance of the method.
Structure of the paper. The paper is structured as follows: in Section 2 we review some of the known results concerning duality for the specific structure of , and we provide a new result related to the primal solutions recovered from the dual. In Section 3 we propose a new method for primal solution recovery, and provide performance bounds for these solutions. We also give some results on how to further improve the solutions’ quality in some special cases. In Section 4 we verify the efficacy of our proposed method on a difficult optimization problem stemming from power systems, and in Section 5 we conclude the paper.
Notation. Given some optimization problem , we denote with its optimal objective and with the performance of the solution with respect to the objective of . For a given set , we denote by its convex hull and by the set of vertices of . With “” we always intend component-wise inequalities (between vectors or matrices), and with we indicate the cartesian product of sets. The support of a vector is the set of indexes of the non-zero elements: , while is the projection of onto the positive orthant, i.e., . For the specific structure of , we use the overbar symbol to indicate quantities related to the contracted version of , as introduced in Section 3. Thus, for instance, is the contracted form of and is its dual. We use parenthesis to avoid confusing the sub- and superscripts, e.g., we denote by the part of related to subproblem of problem . Finally, we use the superscript to denote the th row of matrix .
Section snippets
Duality for problem
Consider the dual function of problem , defined as and then associate to this function the optimization problem We call the dual problem of , and we refer collectively to the minimizations within , i.e., as the inner problem. There is substantial practical interest in understanding the properties of the solutions to the inner problem (1) because they are obtained by solving
A distributed solution method for
Theorem 2.5 says that the inner solutions differ from in at most subsystems, where is the dimension of the coupling constraint. Since is feasible with respect to the coupling constraints and attains a better objective than , one can expect the solutions obtained by solving the dual to be nearly feasible and to attain good objective values. In this section we exploit this result to propose a method aimed at obtaining “good” feasible solutions to problem in a distributed
Application example: Charging of Plug-in Electric Vehicles (PEVs)
We consider a fleet of Plug-in (Hybrid) Electric Vehicles (PEVs) that must be charged by drawing power from the same electricity distribution network. As the number of PEVs increases, it becomes necessary to manage their charging pattern in order to avoid excessive stresses on the lines and transformers of the network. The role of interfacing the fleet of PEVs with the network operators is taken over by a so-called aggregator.
In this section we take the perspective of such an aggregator.
Conclusion
We have provided new results concerning the primal solutions recovered from Lagrangian duals of problems structured as . These results are of direct practical interest, in particular if one wishes to distribute the computational burden of calculating solutions to very large instances of such mixed integer programs. The strength of our results lies in the generality of , which can include very sophisticated local models and therefore accommodate a large variety of practical applications.
It
Robin Vujanic received B.Sc and M.Sc degrees in Mechanical Engineering from ETH Zurich in September 2006 and October 2009, respectively. He joined the Automatic Control Laboratory at department of Electrical Engineering and Information Technology at ETH Zurich in November 2009 and obtained the Ph.D. degree in October 2014. He held a postdoctoral position at ETH Zurich until April 2015, when he joined the Australian Centre for Field Robotics at the University of Sydney as a postdoctoral
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Robin Vujanic received B.Sc and M.Sc degrees in Mechanical Engineering from ETH Zurich in September 2006 and October 2009, respectively. He joined the Automatic Control Laboratory at department of Electrical Engineering and Information Technology at ETH Zurich in November 2009 and obtained the Ph.D. degree in October 2014. He held a postdoctoral position at ETH Zurich until April 2015, when he joined the Australian Centre for Field Robotics at the University of Sydney as a postdoctoral researcher, dually funded by a grant from the Swiss National Foundation and the University of Sydney. His research interests include applications of optimization in control and automation, decomposition methods for large scale optimization models and robust optimization approaches for decision making under uncertainty.
Peyman Mohajerin Esfahani holds a dual-appointment as a postdoctoral researcher at the Risk Analytics and Optimization Chair at EPFL and the Automatic Control Laboratory at ETH Zurich since April 2014. He also was a visiting research scholar at the Laboratory for Information and Decision Systems at MIT in 2015. He received the B.Sc. and M.Sc degrees in Electrical Engineering both from Sharif University of Technology, Tehran, Iran, in September 2005 and December 2007, respectively. In June 2008 he joined the Automatic Control Laboratory at ETH Zurich, where he obtained the Ph.D. degree in January 2014. His research interests include theoretical and practical aspects of decision making problems in uncertain and dynamic environments subject to information constraints, with applications to control and security of large-scale and distributed systems.
Paul J. Goulart received B.Sc and M.Sc degrees in Aeronautics and Astronautics from the Massachusetts Institute of Technology (MIT). He was selected as a Gates Scholar at the University of Cambridge, where he received a Ph.D. in Control Engineering in 2007. From 2007 to 2011 he was a Lecturer in control systems in the Department of Aeronautics at Imperial College London, and from 2011–2014 a Senior Researcher in the Automatic Control Laboratory at ETH Zurich. He is currently an Associate Professor in the Department of Engineering Science and Tutorial Fellow of St. Edmund Hall at the University of Oxford. His research interests include model predictive control, robust optimization and control of fluid flows.
Sébastien Mariéthoz received the M.Sc. in electrical engineering and the Ph.D. degree for his work on asymmetrical multilevel converters from EPFL, Lausanne, Switzerland in 1997 and 2005. In 2005 and 2006, he was research fellow with the Power Electronics and Motion Control (PEMC) group of the University of Nottingham, UK, focusing on the design and control of new matrix converter topologies. Between 2006 and 2014, he was senior researcher at the Automatic Control Laboratory of ETH Zürich, Switzerland. Since 2014, he is professor at the Bern University of Applied Sciences, where his research interests focus on the optimal design and control of power electronic systems, new power converter topologies and sensorless control techniques.
Manfred Morari was head of the Department of Information Technology and Electrical Engineering at ETH Zurich from 2009 to 2012 and head of the Automatic Control Laboratory from 1994 to 2008. Before that he was the McCollum–Corcoran Professor of Chemical Engineering and Executive Officer for Control and Dynamical Systems at the California Institute of Technology. From 1977 to 1983 he was on the faculty of the University of Wisconsin. He obtained the diploma from ETH Zurich and the Ph.D. from the University of Minnesota, both in chemical engineering.
His interests are in hybrid systems and the control of biomedical systems. Morari’s research is internationally recognized. The analysis techniques and software developed in his group are used in universities and industry throughout the world. He has received numerous awards, including the Eckman Award, Ragazzini Award and Bellman Control Heritage Award from the American Automatic Control Council; the Colburn Award, Professional Progress Award and CAST Division Award from the American Institute of Chemical Engineers; the Control Systems Technical Field Award and the Bode Lecture Prize from IEEE. He is a Fellow of IEEE, AIChE and IFAC. In 1993 he was elected to the U.S. National Academy of Engineering. Manfred Morari served on the technical advisory boards of several major corporations.
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The material in this paper was partially presented at the 53rd IEEE Conference on Decision and Control, December 15–17, 2014, Los Angeles, CA, USA and at the 22nd Mediterranean Conference on Control and Automation, June 16–19, 2014, Palermo, Italy. This paper was recommended for publication in revised form by Editor Berç Rüstem.