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Reformulations in mathematical programming: Symmetry
 Mathematical Programming
, 2009
"... If a mathematical program (be it linear or nonlinear) has many symmetric optima, solving it via BranchandBound techniques often yields search trees of disproportionate sizes; thus, finding and exploiting symmetries is an important task. We propose a method for: (a) automatically finding the formul ..."
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If a mathematical program (be it linear or nonlinear) has many symmetric optima, solving it via BranchandBound techniques often yields search trees of disproportionate sizes; thus, finding and exploiting symmetries is an important task. We propose a method for: (a) automatically finding the formulation group of any given MixedInteger Nonlinear Program, and (b) reformulating the problem so that some symmetric solutions become infeasible. The reformulated problem can then be solved via standard BranchandBound codes such as CPLEX (for linear programs) and Couenne (for nonlinear programs). Our computational results include formulation group tables for the MIPLib3, MIPLib2003, GlobalLib and MINLPLib instance libraries, solution tables for some instances in the aforementioned libraries, and a theoretical and computational study of the symmetries of the Kissing Number Problem. 1
Feasibilitybased bounds tightening via fixed points
"... Abstract. The search tree size of the spatial BranchandBound algorithm for MixedInteger Nonlinear Programming depends on many factors, one of which is the width of the variable ranges at every tree node. A range reduction technique often employed is called Feasibility Based Bounds Tightening, whi ..."
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Abstract. The search tree size of the spatial BranchandBound algorithm for MixedInteger Nonlinear Programming depends on many factors, one of which is the width of the variable ranges at every tree node. A range reduction technique often employed is called Feasibility Based Bounds Tightening, which is known to be practically fast, and is thus deployed at every node of the search tree. From time to time, however, this technique fails to converge to its limit point in finite time, thereby slowing the whole BranchandBound search considerably. In this paper we propose a polynomial time method, based on solving a linear program, for computing the limit point of the Feasibility Based Bounds Tightening algorithm applied to linear equality and inequality constraints. Keywords: global optimization, MINLP, spatial BranchandBound, range reduction, constraint programming. 1
A Storm of Feasibility Pumps for Nonconvex MINLP
, 2010
"... One of the foremost difficulties in solving Mixed Integer Nonlinear Programs, either with exact or heuristic methods, is to find a feasible point. We address this issue with a new feasibility pump algorithm tailored for nonconvex Mixed Integer Nonlinear Programs. Feasibility pumps are successive pro ..."
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One of the foremost difficulties in solving Mixed Integer Nonlinear Programs, either with exact or heuristic methods, is to find a feasible point. We address this issue with a new feasibility pump algorithm tailored for nonconvex Mixed Integer Nonlinear Programs. Feasibility pumps are successive projection algorithms that iterate between solving a continuous relaxation and a mixedinteger relaxation of the original problems; such approaches currently exist in the literature for MixedInteger Linear Programs and convex MixedInteger Nonlinear Programs. Both cases exhibit the distinctive property that the continuous relaxation can be solved in polynomial time. In nonconvex Mixed Integer Nonlinear Programming such a property does not hold and the main innovations in this paper are tailored algorithmic methods to overcome such a difficulty. We present extensive computational results on the MINLPLib, showing the effectiveness and efficiency of our algorithm.
On Intervalsubgradient and Nogood Cuts
 OPERATIONS RESEARCH LETTERS
, 2010
"... Intervalgradient cuts are (nonlinear) valid inequalities for nonconvex NLPs defined for constraints g(x) ≤ 0 with g being continuously differentiable in a box [x, ¯x]. In this paper we define intervalsubgradient cuts, a generalization to the case of nondifferentiable g, and show that nogood cuts ..."
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Intervalgradient cuts are (nonlinear) valid inequalities for nonconvex NLPs defined for constraints g(x) ≤ 0 with g being continuously differentiable in a box [x, ¯x]. In this paper we define intervalsubgradient cuts, a generalization to the case of nondifferentiable g, and show that nogood cuts (which have the form ‖x−ˆx ‖ ≥ ε for some norm and positive constant ε) are a special case of intervalsubgradient cuts whenever the 1norm is used. We then briefly discuss what happens if other norms are used.
Constraint and Optimization techniques for supporting Policy Making
"... Modeling the policy making process is a very challenging task. To the best of our knowledge the most widely used technique in this setting is agentbased simulation. Each agent represents an individual entity (e.g., citizen, stakeholder, company, public association, public body). The agent interacti ..."
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Modeling the policy making process is a very challenging task. To the best of our knowledge the most widely used technique in this setting is agentbased simulation. Each agent represents an individual entity (e.g., citizen, stakeholder, company, public association, public body). The agent interaction enables emerging behaviours to be observed and taken into account in the policy making process itself. We claim that another perspective should be considered in modeling policy issues, that is the global perspective. Each public body has global objectives, constraints and guidelines that have to be combined to take decisions. The policy making process should be at the same time consistent with constraints, optimal with respect to given objectives and assessed to avoid negative impacts on the environment, economy and
Contents 1 Constraint and Optimization techniques for supporting Policy Making 3
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Computational Management Science manuscript No. (will be inserted by the editor) Optimization and Sustainable Development
, 2014
"... Abstract In this opinion paper, I argue that “optimization and sustainable development” indicates a set of specific engineering techniques rather than a unified discipline stemming from a unique scientific principle. On the other hand, I also propose a mathematical principle underlying at least some ..."
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Abstract In this opinion paper, I argue that “optimization and sustainable development” indicates a set of specific engineering techniques rather than a unified discipline stemming from a unique scientific principle. On the other hand, I also propose a mathematical principle underlying at least some of the concepts defining sustainability when optimizing a supply chain. The principle is based on the fact that since demand constraints are usually expressed as inequalities, those which are not active at the optimal solution imply the existence of some wasted activity, which may lead to an unsustainable solution. I propose using flowtype equation constraints instead, which help detect unsustainability through infeasibility.