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64
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 39 (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.
Optimal MultiChannel Cooperative Sensing in Cognitive Radio Networks
 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 9
, 2010
"... Abstract—In this paper, optimal multichannel cooperative sensing strategies in cognitive radio networks are investigated. A cognitive radio network with multiple potential channels is considered. Secondary users cooperatively sense the channels and send the sensing results to a coordinator, in whic ..."
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Cited by 31 (2 self)
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Abstract—In this paper, optimal multichannel cooperative sensing strategies in cognitive radio networks are investigated. A cognitive radio network with multiple potential channels is considered. Secondary users cooperatively sense the channels and send the sensing results to a coordinator, in which energy detection with a soft decision rule is employed to estimate whether there are primary activities in the channels. An optimization problem is formulated, which maximizes the throughput of secondary users while keeping detection probability for each channel above a predefined threshold. In particular, two sensing modes are investigated: slottedtime sensing mode and continuoustime sensing mode. With a slottedtime sensing mode, the sensing time of each secondary user consists of a number of minislots, each of which can be used to sense one channel. The initial optimization problem is shown to be a nonconvex mixedinteger problem. A polynomialcomplexity algorithm is proposed to solve the problem optimally. With a continuoustime sensing mode, the sensing time of each secondary user for a channel can be any arbitrary continuous value. The initial nonconvex problem is converted into a convex bilevel problem, which can be successfully solved by existing methods. Numerical results are presented to demonstrate the effectiveness of our proposed algorithms. Index Terms—Cognitive radio, spectrum sensing, throughput maximization. I.
A StiffnessBased Quality Measure for Compliant Grasps and Fixtures
, 2000
"... This paper presents a systematic approach to quantifying the effectiveness of compliant grasps and fixtures of an object. The approach is physically motivated and applies to the grasping of 2D and 3D objects by any number of fingers. The approach is based on a characterization of the frameinvariant ..."
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Cited by 26 (0 self)
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This paper presents a systematic approach to quantifying the effectiveness of compliant grasps and fixtures of an object. The approach is physically motivated and applies to the grasping of 2D and 3D objects by any number of fingers. The approach is based on a characterization of the frameinvariant features of a grasp or fixture stiffness matrix. In particular, we define a set of frameinvariant characteristic stiffness parameters, and provide physical and geometric interpretation for these parameters. Using a physically meaningful scheme to make the rotational and translational stiffness parameters comparable, we define a frameinvariant quality measure, which we call the stiffness quality measure. An example of a frictional grasp illustrates the effectiveness of the quality measure. We then consider the optimal grasping of frictionless polygonal objects by three and four fingers. Such frictionless grasps are useful in highload fixturing applications, and their relative simplicity allows ...
Nonnegative Sparse PCA
 In Neural Information Processing Systems
, 2007
"... We describe a nonnegative variant of the ”Sparse PCA ” problem. The goal is to create a low dimensional representation from a collection of points which on the one hand maximizes the variance of the projected points and on the other uses only parts of the original coordinates, and thereby creating a ..."
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Cited by 26 (1 self)
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We describe a nonnegative variant of the ”Sparse PCA ” problem. The goal is to create a low dimensional representation from a collection of points which on the one hand maximizes the variance of the projected points and on the other uses only parts of the original coordinates, and thereby creating a sparse representation. What distinguishes our problem from other Sparse PCA formulations is that the projection involves only nonnegative weights of the original coordinates — a desired quality in various fields, including economics, bioinformatics and computer vision. Adding nonnegativity contributes to sparseness, where it enforces a partitioning of the original coordinates among the new axes. We describe a simple yet efficient iterative coordinatedescent type of scheme which converges to a local optimum of our optimization criteria, giving good results on large real world datasets. 1
PERFECT DUALITY THEORY AND COMPLETE SOLUTIONS TO A CLASS OF GLOBAL OPTIMIZATION PROBLEMS
, 2003
"... This article presents a complete set of solutions for a class of global optimization problems. These problems are directly related to numericalization of a large class of semilinear nonconvex partial differential equations in nonconvex mechanics including phase transitions, chaotic dynamics, nonline ..."
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Cited by 24 (15 self)
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This article presents a complete set of solutions for a class of global optimization problems. These problems are directly related to numericalization of a large class of semilinear nonconvex partial differential equations in nonconvex mechanics including phase transitions, chaotic dynamics, nonlinear field theory, and superconductivity. The method used is the socalled canonical dual transformation developed recently. It is shown that, by this method, these difficult nonconvex constrained primal problems in R n can be converted into a onedimensional canonical dual problem, i.e. the perfect dual formulation with zero duality gap and without any perturbation. This dual criticality condition leads to an algebraic equation which can be solved completely. Therefore, a complete set of solutions to the primal problems is obtained. The extremality of these solutions are controlled by the triality theory discovered recently [D.Y. Gao (2000). Duality Principles in Nonconvex Systems: Theory, Methods and Applications, Vol. xviii, p. 454. Kluwer Academic Publishers, Dordrecht/Boston/London.]. Several examples are illustrated including the nonconvex constrained quadratic programming. Results show that these problems can be solved completely to obtain all KKT points and global minimizers.
Canonical dual approach for solving 01 quadratic programming problems
 J. Industrial and Management Optimization
, 2007
"... Abstract. By using the canonical dual transformation developed recently, we derive a pair of canonical dual problems for 01 quadratic programming problems in both minimization and maximization form. Regardless convexity, when the canonical duals are solvable, no duality gap exists between the prima ..."
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Cited by 23 (12 self)
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Abstract. By using the canonical dual transformation developed recently, we derive a pair of canonical dual problems for 01 quadratic programming problems in both minimization and maximization form. Regardless convexity, when the canonical duals are solvable, no duality gap exists between the primal and corresponding dual problems. Both global and local optimality conditions are given. An algorithm is presented for finding global minimizers, even when the primal objective function is not convex. Examples are included to illustrate this new approach.
Reformulations in Mathematical Programming: A Computational Approach
"... Mathematical programming is a language for describing optimization problems; it is based on parameters, decision variables, objective function(s) subject to various types of constraints. The present treatment is concerned with the case when objective(s) and constraints are algebraic mathematical ex ..."
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Cited by 21 (17 self)
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Mathematical programming is a language for describing optimization problems; it is based on parameters, decision variables, objective function(s) subject to various types of constraints. The present treatment is concerned with the case when objective(s) and constraints are algebraic mathematical expressions of the parameters and decision variables, and therefore excludes optimization of blackbox functions. A reformulation of a mathematical program P is a mathematical program Q obtained from P via symbolic transformations applied to the sets of variables, objectives and constraints. We present a survey of existing reformulations interpreted along these lines, some example applications, and describe the implementation of a software framework for reformulation and optimization.
Global Search Methods For Solving Nonlinear Optimization Problems
, 1997
"... ... these new methods, we develop a prototype, called Novel (Nonlinear Optimization Via External Lead), that solves nonlinear constrained and unconstrained problems in a unified framework. We show experimental results in applying Novel to solve nonlinear optimization problems, including (a) the lear ..."
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Cited by 18 (1 self)
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... these new methods, we develop a prototype, called Novel (Nonlinear Optimization Via External Lead), that solves nonlinear constrained and unconstrained problems in a unified framework. We show experimental results in applying Novel to solve nonlinear optimization problems, including (a) the learning of feedforward neural networks, (b) the design of quadraturemirrorfilter digital filter banks, (c) the satisfiability problem, (d) the maximum satisfiability problem, and (e) the design of multiplierless quadraturemirrorfilter digital filter banks. Our method achieves better solutions than existing methods, or achieves solutions of the same quality but at a lower cost.
2007) Scatter Search for chemical and bioprocess optimization
 Journal of Global Optimization
"... Scatter search is a populationbased method that has recently been shown to yield promising outcomes for solving combinatorial and nonlinear optimization problems. Based on formulations originally proposed in the 1960s for combining decision rules and problem constraints such as the surrogate constr ..."
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Cited by 17 (3 self)
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Scatter search is a populationbased method that has recently been shown to yield promising outcomes for solving combinatorial and nonlinear optimization problems. Based on formulations originally proposed in the 1960s for combining decision rules and problem constraints such as the surrogate constraint method, scatter search uses strategies for combining solution vectors that have proved effective in a variety of problem settings. In this paper, we develop a general purpose heuristic for a class of nonlinear optimization problems. The procedure is based on the scatter search methodology and treats the objective function evaluation as a black box, making the search algorithm contextindependent. Most optimization problems in the chemical and biochemical industries are highly nonlinear in either the objective function or the constraints. Moreover, they usually present differentialalgebraic systems of constraints. In this type of problem, the evaluation of a solution or even the feasibility test of a set of values for the decision variables is a timeconsuming operation. In this context, the solution method is limited to a reduced number of solution examinations. We have implemented a scatter search procedure in Matlab for this special class of difficult optimization problems. Our development goes beyond a simple exercise of applying scatter search to this class of problem, but presents innovative mechanisms to obtain a good balance between intensification and diversification in a shortterm search horizon. Computational comparisons with other recent methods over a set of benchmark problems favor the proposed procedure.
Canonical Duality Theory and Solutions to Constrained Nonconvex Quadratic Programming
, 2004
"... This paper presents a perfect duality theory and a complete set of solutions to nonconvex quadratic programming problems subjected to inequality constraints. By use of the canonical dual transformation developed recently, a canonical dual problem is formulated, which is perfectly dual to the primal ..."
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Cited by 14 (7 self)
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This paper presents a perfect duality theory and a complete set of solutions to nonconvex quadratic programming problems subjected to inequality constraints. By use of the canonical dual transformation developed recently, a canonical dual problem is formulated, which is perfectly dual to the primal problem in the sense that they have the same set of KKT points. It is proved that the KKT points depend on the index of the Hessian matrix of the total cost function. The global and local extrema of the nonconvex quadratic function can be identified by the triality theory [11]. Results show that if the global extrema of the nonconvex quadratic function are located on the boundary of the primal feasible space, the dual solutions should be interior points of the dual feasible set, which can be solved by deterministic methods. Certain nonconvex quadratic programming problems in � n can be converted into a dual problem with only one variable. It turns out that a complete set of solutions for quadratic programming over a sphere is obtained as a byproduct. Several examples are illustrated.