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50
Global minimization using an Augmented Lagrangian method with variable lowerlevel constraints
, 2007
"... A novel global optimization method based on an Augmented Lagrangian framework is introduced for continuous constrained nonlinear optimization problems. At each outer iteration k the method requires the εkglobal minimization of the Augmented Lagrangian with simple constraints, where εk → ε. Global c ..."
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Cited by 21 (1 self)
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A novel global optimization method based on an Augmented Lagrangian framework is introduced for continuous constrained nonlinear optimization problems. At each outer iteration k the method requires the εkglobal minimization of the Augmented Lagrangian with simple constraints, where εk → ε. Global convergence to an εglobal minimizer of the original problem is proved. The subproblems are solved using the αBB method. Numerical experiments are presented.
Riskaverse strategies for security games with execution and observational uncertainty
 In AAAI
, 2011
"... Attackerdefender Stackelberg games have become a popular gametheoretic approach for security with deployments for LAX Police, the FAMS and the TSA. Unfortunately, most of the existing solution approaches do not model two key uncertainties of the realworld: there may be noise in the defender’s exe ..."
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Cited by 12 (11 self)
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Attackerdefender Stackelberg games have become a popular gametheoretic approach for security with deployments for LAX Police, the FAMS and the TSA. Unfortunately, most of the existing solution approaches do not model two key uncertainties of the realworld: there may be noise in the defender’s execution of the suggested mixed strategy and/or the observations made by an attacker can be noisy. In this paper, we provide a framework to model these uncertainties, and demonstrate that previous strategies perform poorly in such uncertain settings. We also provide RECON, a novel algorithm that computes strategies for the defender that are robust to such uncertainties, and provide heuristics that further improve RECON’s efficiency.
Modeling Pilot's Sequential Maneuvering Decisions by a Multistage Influence Diagram
 Journal of Guidance, Control, and Dynamics
, 2001
"... We present a modeling and analysis approach that offers a way to generate preference optimal flight paths in oneonone air combat. The pilot's sequential maneuvering decisions are modeled by a multistage influence diagram. The influence diagram graphically describes the elements of the decision pro ..."
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Cited by 9 (4 self)
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We present a modeling and analysis approach that offers a way to generate preference optimal flight paths in oneonone air combat. The pilot's sequential maneuvering decisions are modeled by a multistage influence diagram. The influence diagram graphically describes the elements of the decision process, contains a pointmass model for the dynamics of an aircraft and takes into account the decision maker's preferences under conditions of uncertainty. Optimal trajectories with respect to the given preference model are obtained by converting the multistage influence diagram into a discrete time dynamic optimization problem that is solved by nonlinear programming. The presented approach is illustrated by analyzing two oneonone air combat scenarios.
A Mathematical Model and Descent Algorithm for Bilevel Traffic Management
, 2002
"... We provide a new mathematical model for strategic traffic management, formulated and analyzed as a mathematical program with equilibrium constraints (MPEC). The model includes two types of control (upperlevel) variables, which may be used to describe such traffic management actions as traffic signa ..."
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Cited by 9 (4 self)
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We provide a new mathematical model for strategic traffic management, formulated and analyzed as a mathematical program with equilibrium constraints (MPEC). The model includes two types of control (upperlevel) variables, which may be used to describe such traffic management actions as traffic signal setting, network design, and congestion pricing. The lowerlevel problem of the MPEC describes a traffic equilibrium model in the sense of Wardrop, in which the control variables enter as parameters in the travel costs. We consider a (small) variety of model settings, including fixed or elastic demands, the possible presence of side constraints in the traffic equilibrium system, and representations of traffic flows and management actions in both linkroute and linknode space. For this model, we also propose and analyze a descent algorithm. The algorithm utilizes a new reformulation of the MPEC into a constrained, locally Lipschitz minimization problem in the product space of controls and traffic flows. The reformulation is based on the Minty (1967) parameterization of the graph of the normal cone operator for the traffic flow polyhedron. Two immediate advantages of making use of this reformulation are that the resulting descent algorithm can be operated and established to be convergent without requiring that the travel cost mapping is monotone, and without having to ever solve the lowerlevel equilibrium problem. We provide example realizations of the algorithm, establish their convergence, and interpret their workings in terms of the traffic network.
New necessary optimality conditions for bilevel programs by combining MPEC and the value function approach
 SIAM J. Optim
"... Abstract. The bilevel program is a sequence of two optimization problems where the constraint region of the upper level problem is determined implicitly by the solution set to the lower level problem. The classical approach to solving such a problem is to replace the lower level problem by its Karus ..."
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Cited by 8 (3 self)
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Abstract. The bilevel program is a sequence of two optimization problems where the constraint region of the upper level problem is determined implicitly by the solution set to the lower level problem. The classical approach to solving such a problem is to replace the lower level problem by its Karush–Kuhn–Tucker (KKT) condition and solve the resulting mathematical programming problem with equilibrium constraints (MPEC). In general the classical approach is not valid for nonconvex bilevel programming problems. The value function approach uses the value function of the lower level problem to define an equivalent single level problem. But the resulting problem requires a strong assumption, such as the partial calmness condition, for the KKT condition to hold. In this paper we combine the classical and the value function approaches to derive new necessary optimality conditions under rather weak conditions. The required conditions are even weaker in the case where the classical approach or the value function approach alone is applicable.
Bilevel programming a special Case of a Mathematical Program with Complementarity Constraints?
, 2009
"... Bilevel programming problems are often reformulated using the KarushKuhnTucker conditions for the lower level problem resulting in a mathematical program with complementarity constraints(MPCC). Clearly, both problems are closely related. But the answer to the question posed is “No ” even in the c ..."
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Cited by 7 (3 self)
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Bilevel programming problems are often reformulated using the KarushKuhnTucker conditions for the lower level problem resulting in a mathematical program with complementarity constraints(MPCC). Clearly, both problems are closely related. But the answer to the question posed is “No ” even in the case when the lower level programming problem is a parametric convex optimization problem. This is not obvious and concerns local optimal solutions. We show that global optimal solutions of the MPCC correspond to global optimal solutions of the bilevel problem provided the lowerlevel problem satisfies the Slater’s constraint qualification. We also show by examples that this correspondence can fail if the Slater’s constraint qualification fails to hold at lowerlevel. When we consider the local solutions, the relationship between the bilevel problem and its corresponding MPCC is more complicated. We also demonstrate the issues relating to a local minimum through examples.
Survivable Network Design Under Optimal and Heuristic Interdiction Scenarios
"... Abstract: We examine the problem of building or fortifying a network to defend against enemy attacks in various scenarios. In particular, we examine the case in which an enemy can destroy any portion of any arc that a designer constructs on the network, subject to some interdiction budget. This prob ..."
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Cited by 6 (2 self)
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Abstract: We examine the problem of building or fortifying a network to defend against enemy attacks in various scenarios. In particular, we examine the case in which an enemy can destroy any portion of any arc that a designer constructs on the network, subject to some interdiction budget. This problem takes the form of a threelevel, twoplayer game, in which the designer acts first to construct a network and transmit an initial set of flows through the network. The enemy acts next to destroy a set of constructed arcs in the designer's network, and the designer acts last to transmit a final set of flows in the network. Most studies of this nature assume that the enemy will act optimally; however, in realworld scenarios one cannot necessarily assume rationality on the part of the enemy. Hence, we prescribe optimal network design algorithms for three different profiles of enemy action: an enemy destroying arcs based on capacities, based on initial flows, or acting optimally to minimize our maximum profits obtained from transmitting flows.
A Multicriteria Approach to Bilevel Optimization
"... In this paper we study the relationship between bilevel optimization and multicriteria optimization. Given a bilevel optimization problem, we introduce an order relation such that the optimal solutions of the bilevel problem are the nondominated points with respect to the order relation. In the ca ..."
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Cited by 6 (2 self)
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In this paper we study the relationship between bilevel optimization and multicriteria optimization. Given a bilevel optimization problem, we introduce an order relation such that the optimal solutions of the bilevel problem are the nondominated points with respect to the order relation. In the case where the lower level problem of the bilevel optimization problem is convex and continuously dierentiable in the lower level variables, this order relation is equivalent to a second, more tractable order relation. Then, we show how to construct a (nonconvex) cone for which we can prove that the nondominated points with respect to the order relation induced by the cone are also nondominated points with respect to any of the two order relations mentioned before. We also comment on the practical and computational implications of our approach.
Constraint qualifications and KKT conditions for bilevel programming problems
 Math. Oper. Res
"... doi 10.1287/moor.1060.0219 ..."
Linear Bilevel Programming With Upper Level Constraints Depending on the Lower Level Solution
, 2005
"... Focus in the paper is on the definition of linear bilevel programming problems, the existence of optimal solutions and necessary as well as sufficient optimality conditions. In the papers [9] and [10] the authors claim to suggest a refined definition of linear bilevel programming problems and relate ..."
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Cited by 3 (3 self)
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Focus in the paper is on the definition of linear bilevel programming problems, the existence of optimal solutions and necessary as well as sufficient optimality conditions. In the papers [9] and [10] the authors claim to suggest a refined definition of linear bilevel programming problems and related optimality conditions. Mainly their attempt reduces to shifting upper level constraints involving both the upper and the lower level variables into the lower level. We investigate such a shift in more details and show that it is not allowed in general. We show that an optimal solution of the bilevel programm exists under the conditions in [10] if we add the assumption that the inducible region is not empty. The necessary optimality condition reduces to check optimality in one linear programming problem. Optimality of one feasible point for a certain number of linear programs implies optimality for the bilevel problem. 1