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The zero duality gap property and lower semicontinuity of the perturbation function
"... We examine the validity of the zero duality gap properties for two important dual schemes: a generalized augmented Lagrangian dual scheme and a nonlinear Lagrangetype dual scheme. The necessary and su#cient conditions for the zero duality gap property to hold are established in terms of the lower se ..."
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Cited by 11 (4 self)
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We examine the validity of the zero duality gap properties for two important dual schemes: a generalized augmented Lagrangian dual scheme and a nonlinear Lagrangetype dual scheme. The necessary and su#cient conditions for the zero duality gap property to hold are established in terms of the lower semicontinuity of the perturbation functions.
On a modified subgradient algorithm for dual problems via sharp augmented Lagrangian
 Journal of Global Optimization
, 2006
"... We study convergence properties of a modified subgradient algorithm, applied to the dual problem defined by the sharp augmented Lagrangian. The primal problem we consider is nonconvex and nondifferentiable, with equality constraints. We obtain primal and dual convergence results, as well as a condit ..."
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Cited by 8 (2 self)
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We study convergence properties of a modified subgradient algorithm, applied to the dual problem defined by the sharp augmented Lagrangian. The primal problem we consider is nonconvex and nondifferentiable, with equality constraints. We obtain primal and dual convergence results, as well as a condition for existence of a dual solution. Using a practical selection of the stepsize parameters, we demonstrate the algorithm and its advantages on test problems, including an integer programming and an optimal control problem. Key words: Nonconvex programming; nonsmooth optimization; augmented Lagrangian; sharp Lagrangian; subgradient optimization.
Extended duality for nonlinear programming
, 2008
"... Duality is an important notion for nonlinear programming (NLP). It provides a theoretical foundation for many optimization algorithms. Duality can be used to directly solve NLPs as well as to derive lower bounds of the solution quality which have wide use in other highlevel search techniques such a ..."
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Cited by 5 (2 self)
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Duality is an important notion for nonlinear programming (NLP). It provides a theoretical foundation for many optimization algorithms. Duality can be used to directly solve NLPs as well as to derive lower bounds of the solution quality which have wide use in other highlevel search techniques such as branch and bound. However, the conventional duality theory has the fundamental limit that it leads to duality gaps for nonconvex problems, including discrete and mixedinteger problems where the feasible sets are generally nonconvex. In this paper, we propose an extended duality theory for nonlinear optimization in order to overcome some limitations of previous dual methods. Based on a new dual function, the extended duality theory leads to zero duality gap for general nonconvex problems defined in discrete, continuous, and mixed spaces under mild conditions. Comparing to recent developments in nonlinear Lagrangian functions and exact penalty functions, the proposed theory always requires lesser penalty to achieve zero duality. This is very desirable as the lower function value leads to smoother search terrains and alleviates the ill conditioning of dual optimization. Based on the extended duality theory, we develop a general search framework for global optimization. Experimental results on engineering benchmarks and a sensornetwork optimization application show that our algorithm achieves better performance than searches based on conventional duality and Lagrangian theory.
A Duality Approach to Path Planning for Multiple Robots †
"... Abstract — In this paper, we propose an optimizationbased framework for path planning for multiple robots in presence of obstacles. The objective is to find multiple fixed length paths for multiple robots that satisfy the following constraints: (i) bounded curvature, (ii) obstacle avoidance, (iii) ..."
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Abstract — In this paper, we propose an optimizationbased framework for path planning for multiple robots in presence of obstacles. The objective is to find multiple fixed length paths for multiple robots that satisfy the following constraints: (i) bounded curvature, (ii) obstacle avoidance, (iii) and collision avoidance. First, we formulate a relaxation of the path planning problem using polygonal approximations. We show that path planning problem for multiple robots under various constraints and missions, such as curvature and obstacle avoidance constraints as well as rendezvous and maximal total area coverage, can be cast as a nonconvex optimization problem. Then, we propose an alternative dual formulation that results in no duality gap. We show that the alternative dual function can be interpreted as minimum potential energy of a multiparticle system with discontinuous springlike forces. Finally, we show that using the proposed dualitybased framework, an approximation of the minimal length path planning problem (also known as Dubins ’ problem) in presence of obstacles can be solved efficiently using primaldual interiorpoint methods. I.
Strictly Increasing Positively Homogeneous Functions with Application to Exact Penalization
"... We study a nonlinear exact penalization for optimization problems with a single constraint. The penalty function is constructed as a convolution of the objective function and the constraint by means of IPH (increasing positively homogeneous) functions. The main results are obtained for penalization ..."
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We study a nonlinear exact penalization for optimization problems with a single constraint. The penalty function is constructed as a convolution of the objective function and the constraint by means of IPH (increasing positively homogeneous) functions. The main results are obtained for penalization by strictly IPH functions. We show that some restrictive assumptions, which have been made in earlier researches on this topic, can be removed. We also compare the least exact penalty parameters for penalization by di#erent convolution functions. These results are based on some properties of strictly IPH functions that are established in the paper.
WeC01.4 Path Planning for Multiple Robots: An Alternative Duality Approach †
, 2010
"... Abstract — We propose an optimizationbased framework to find multiple fixed length paths for multiple robots that satisfy the following constraints: (i) bounded curvature, (ii) obstacle avoidance, (iii) and collision avoidance. By using polygonal approximation techniques, we show that path planning ..."
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Abstract — We propose an optimizationbased framework to find multiple fixed length paths for multiple robots that satisfy the following constraints: (i) bounded curvature, (ii) obstacle avoidance, (iii) and collision avoidance. By using polygonal approximation techniques, we show that path planning problem for multiple robots under various constraints and missions, such as curvature and obstacle avoidance constraints as well as rendezvous and maximal total area coverage, can be cast as a nonconvex optimization problem. Then, we propose an alternative dual formulation that results in no duality gap. We show that the alternative dual function can be interpreted as minimum potential energy of a multiparticle system with discontinuous springlike forces. Finally, we show that using the proposed dualitybased framework, an approximation of the minimal length path planning problem (also known as Dubins’ problem) in presence of obstacles can be solved efficiently using primaldual interiorpoint methods. I.
DOI 10.1007/s1058900892083 Extended duality for nonlinear programming
, 2007
"... Abstract Duality is an important notion for nonlinear programming (NLP). It provides a theoretical foundation for many optimization algorithms. Duality can be used to directly solve NLPs as well as to derive lower bounds of the solution quality which have wide use in other highlevel search techniqu ..."
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Abstract Duality is an important notion for nonlinear programming (NLP). It provides a theoretical foundation for many optimization algorithms. Duality can be used to directly solve NLPs as well as to derive lower bounds of the solution quality which have wide use in other highlevel search techniques such as branch and bound. However, the conventional duality theory has the fundamental limit that it leads to duality gaps for nonconvex problems, including discrete and mixedinteger problems where the feasible sets are generally nonconvex. In this paper, we propose an extended duality theory for nonlinear optimization in order to overcome some limitations of previous dual methods. Based on a new dual function, the extended duality theory leads to zero duality gap for general nonconvex problems defined in discrete, continuous, and mixed spaces under mild conditions. Comparing to recent developments in nonlinear Lagrangian functions and exact penalty functions, the proposed theory always requires lesser penalty to achieve zero duality. This is very desirable as the lower function value leads to smoother search terrains and alleviates the ill conditioning of dual optimization. Based on the extended duality theory, we develop a general search framework for global optimization. Experimental results on engineering benchmarks and a sensornetwork optimization application show that our algorithm achieves better performance than searches based on conventional duality and Lagrangian theory.
A Deflected Subgradient Method Using a General Augmented Lagrangian Duality With Implications on Penalty Methods
, 2009
"... We propose a duality scheme for solving constrained nonsmooth and nonconvex optimization problems. Our approach is to use a new variant of the deflected subgradient method for solving the dual problem. Our augmented Lagrangian function induces a primaldual method with strong duality, i.e., with z ..."
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We propose a duality scheme for solving constrained nonsmooth and nonconvex optimization problems. Our approach is to use a new variant of the deflected subgradient method for solving the dual problem. Our augmented Lagrangian function induces a primaldual method with strong duality, i.e., with zero duality gap. We prove that our method converges to a dual solution if and only if a dual solution exists. We also prove that all accumulation points of an auxiliary primal sequence are primal solutions. Our results apply, in particular, to classical penalty methods, since the penalty functions associated with these methods can be recovered as a special case of our augmented Lagrangians. Besides the classical augmenting terms given by the 1 or 2norm forms, terms of many other forms can be used in our Lagrangian function. Using a practical selection of the stepsize parameters, as well as various choices of the augmenting term, we demonstrate the method on test problems. Our numerical experiments indicate that it is more favourable to use an augmenting term of an exponential form rather than the classical 1 or 2norm forms.
Nonsmooth Analysis of Spectral Functions
, 2002
"... Any spectral function can be written as a composition function of a symmetric , often denoted by #). ..."
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Any spectral function can be written as a composition function of a symmetric , often denoted by #).
ON GENERAL AUGMENTED LAGRANGIANS AND A MODIFIED SUBGRADIENT ALGORITHM
, 2009
"... In this thesis we study a modified subgradient algorithm applied to the dual problem generated by augmented Lagrangians. We consider an optimization problem with equality constraints and study an exact version of the algorithm with a sharp Lagrangian in finite dimensional spaces. An inexact versi ..."
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In this thesis we study a modified subgradient algorithm applied to the dual problem generated by augmented Lagrangians. We consider an optimization problem with equality constraints and study an exact version of the algorithm with a sharp Lagrangian in finite dimensional spaces. An inexact version of the algorithm is extended to infinite dimensional spaces and we apply it to a dual problem of an extended realvalued optimization problem. The dual problem is constructed via augmented Lagrangians which include sharp Lagrangian as a particular case. The sequences generated by these algorithms converge to a dual solution when the dual optimal solution set is nonempty. They have the property that all accumulation points of a primal sequence, obtained without extra cost, are primal solutions. We relate the convergence properties of these modified subgradient algorithms to differentiability of the dual function at a dual solution, and exact penalty property of these augmented Lagrangians. In the second part of this thesis, we propose and analyze a general augmented Lagrangian function, which includes several augmented Lagrangians considered in the literature. In this more general setting, we study a zero duality gap property, exact penalization and convergence of a suboptimal path related to the dual problem.