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On stringaveraging for sparse problems and on the split common fixed point problem
, 2008
"... We review the common fi
xed point problem for the class of directed operators. This class is important because many commonly used nonlinear operators in convex optimization belong to it. We present our recent de
finition of sparseness of a family of operators and discuss a stringaveraging algorith ..."
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Cited by 9 (9 self)
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We review the common fi
xed point problem for the class of directed operators. This class is important because many commonly used nonlinear operators in convex optimization belong to it. We present our recent de
finition of sparseness of a family of operators and discuss a stringaveraging algorithmic scheme that favorably handles the common fixed points problem when the family of operators is sparse. We also review some recent results on the multiple operators split common fixed point problem which requires to
find a common
xed point of a family of operators in one space whose image under a linear transformation is a common fi
xed point of another family of operators in the image space.
Discovering the Characteristics of Mathematical Programs via Sampling
 Optimization Methods and Software
, 2002
"... Complex models, large scale: – Unexpected results, bad performance, solver failure… Limited information returned by (e.g. NLP) solvers: – Feasible, KKT conditions satisfied – No improvement in many iterations: stopping. – Unable to find feasible point. – Too many iterations. – Various specific failu ..."
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Cited by 8 (1 self)
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Complex models, large scale: – Unexpected results, bad performance, solver failure… Limited information returned by (e.g. NLP) solvers: – Feasible, KKT conditions satisfied – No improvement in many iterations: stopping. – Unable to find feasible point. – Too many iterations. – Various specific failure messages… Questions: – Why do I have this problem? – How do I make the solver run better on this model? Needed: tools to discover the characteristics of models Discovering Characteristics of Math Programs 2Model Characteristics Some characteristics (e.g. for NLPs): • Shapes of the constraints and objective (convex, concave, both, almost linear, etc.) • Shape of the feasible region (convex, nonconvex) • Redundancy of constraints • Location of feasible region Insights gained: • Better understanding of outcomes and behaviour • Functions that can be approximated (e.g. linear) • Constraints that can be ignored • Best type of solution algorithm to apply • Good starting point
On The Behavior of Subgradient Projections Methods for Convex Feasibility Problems in Euclidean Spaces
, 2008
"... We study some methods of subgradient projections for solving a convex feasibility problem with general (not necessarily hyperplanes or halfspaces) convex sets in the inconsistent case and propose a strategy that controls the relaxation parameters in a specific selfadapting 1 manner. This strategy ..."
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Cited by 5 (4 self)
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We study some methods of subgradient projections for solving a convex feasibility problem with general (not necessarily hyperplanes or halfspaces) convex sets in the inconsistent case and propose a strategy that controls the relaxation parameters in a specific selfadapting 1 manner. This strategy leaves enough userflexibility but gives a mathematical guarantee for the algorithm’s behavior in the inconsistent case. We present numerical results of computational experiments that illustrate the computational advantage of the new method. 1
Algorithms for the Quasiconvex Feasibility Problem Yair Censor
, 2004
"... We study the behavior of subgradient projections algorithms for the quasiconvex feasibility problem of finding a point x ∗ ∈ Rn that satisfies the inequalities f1(x∗) ≤ 0, f2(x∗) ≤ 0,..., fm(x∗) ≤ 0, where all functions are continuous and quasiconvex. We consider the consistent case when the so ..."
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Cited by 5 (3 self)
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We study the behavior of subgradient projections algorithms for the quasiconvex feasibility problem of finding a point x ∗ ∈ Rn that satisfies the inequalities f1(x∗) ≤ 0, f2(x∗) ≤ 0,..., fm(x∗) ≤ 0, where all functions are continuous and quasiconvex. We consider the consistent case when the solution set is nonempty. Since the FenchelMoreau subdifferential might be empty we look at different notions of the subdifferential and determine their suitability for our problem. We also determine conditions on the functions, that are needed for convergence of our algorithms. The quasiconvex functions on the lefthand side of the inequalities need not be differentiable but have to satisfy a Lipschitz or a Hölder condition. 1
Twostage path planning approach for solving multiple spacecraft reconfiguration maneuvers
 Journal of Astronautical Sciences
, 2009
"... The paper presents a twostage approach for designing optimal reconfiguration maneuvers for multiple spacecraft in close proximity. These maneuvers involve wellcoordinated and highlycoupled motions of the entire fleet of spacecraft while satisfying an arbitrary number of constraints. This problem ..."
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Cited by 4 (3 self)
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The paper presents a twostage approach for designing optimal reconfiguration maneuvers for multiple spacecraft in close proximity. These maneuvers involve wellcoordinated and highlycoupled motions of the entire fleet of spacecraft while satisfying an arbitrary number of constraints. This problem is complicated by the nonlinearity of the attitude dynamics, the nonconvexity of some of the constraints, and the coupling that exists in some of the constraints between the positions and attitudes of all spacecraft. While there has been significant research to solve for the translation and/or rotation trajectories for the multiple spacecraft reconfiguration problem, the approach presented in this paper is more general and on a larger scale than the problems considered previously. The essential feature of the solution approach is the separation into two stages, the first using a simplified planning approach to obtain a feasible solution, which is then significantly improved using a smoothing stage. The first stage is solved using a bidirectional Rapidlyexploring Random Tree (RRT) planner. Then the second step optimizes the trajectories by solving an optimal control problem using the Gauss pseudospectral method (GPM). Several examples are presented to demonstrate the effectiveness of the approach for designing spacecraft reconfiguration maneuvers.
Sparse stringaveraging and split common fixed points
, 2008
"... We review the common fixed point problem for the class of directed operators. This class is important because many commonly used nonlinear operators in convex optimization belong to it. We present our recent definition of sparseness of a family of operators and discuss a stringaveraging algorithmi ..."
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Cited by 3 (1 self)
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We review the common fixed point problem for the class of directed operators. This class is important because many commonly used nonlinear operators in convex optimization belong to it. We present our recent definition of sparseness of a family of operators and discuss a stringaveraging algorithmic scheme that favorably handles the common fixed points problem when the family of operators is sparse. We also review some recent results on the multiple operators split common fixed point problem which requires to find a common fixed point of a family of operators in one space whose image under a linear transformation is a common fixed point of another family of operators in the image space.
Iterative Projection Methods in Biomedical Inverse Problems
, 2007
"... The convex or quasiconvex feasibility problem and the split feasibility problem in the Euclidean space have many applications in various fields of science and technology, particularly in problems of image reconstruction from projections, in solving the fully discretized inverse problem in radiati ..."
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Cited by 2 (1 self)
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The convex or quasiconvex feasibility problem and the split feasibility problem in the Euclidean space have many applications in various fields of science and technology, particularly in problems of image reconstruction from projections, in solving the fully discretized inverse problem in radiation therapy treatment planning, and in other image processing problems. Solving systems of linear equalities and/or inequalities is one of them. The class of methods, generally called Projection Methods, has witnessed great progress in recent years and its member algorithms have been applied with success to fully discretized models of inverse problems in image reconstruction and image processing, and in intensitymodulated radiation therapy. We introduce the reader to this field by reviewing algorithmic structures and specific algorithms for the convex feasibility problem, the quasiconvex feasibility problem and the split feasibility problem.
Inertial Iteration for Split Common FixedPoint Problem for QuasiNonexpansive Operators
"... Inspired by the note on split common fixedpoint problem for quasinonexpansive operators presented by ..."
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Inspired by the note on split common fixedpoint problem for quasinonexpansive operators presented by
RESEARCH ARTICLE Constraint Consensus Concentration for Identifying Disjoint Feasible Regions in Nonlinear Programs
"... It is usually not known in advance whether a nonlinear set of constraints has zero, one, or multiple feasible regions. Further, if one or more feasible regions exist, their locations are usually unknown. We propose a method for exploring the variable space quickly using Constraint Consensus to ident ..."
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It is usually not known in advance whether a nonlinear set of constraints has zero, one, or multiple feasible regions. Further, if one or more feasible regions exist, their locations are usually unknown. We propose a method for exploring the variable space quickly using Constraint Consensus to identify promising areas that may contain a feasible region. Multiple Constraint Consensus solution points are clustered to identify regions of attraction. A new interpoint distance frequency distribution technique is used to determine the critical distance for the single linkage clustering algorithm, which in turn determines the estimated number of disjoint feasible regions. The effectiveness of multistart global optimization is increased due to better exploration of the variable space, and efficiency is also increased because the expensive local solver is launched just once near each identified feasible region. The method is demonstrated on a variety of highly nonlinear models.
Feasibility and Constraint Analysis of Sets of Linear Matrix Inequalities
"... We present a constraint analysis methodology for Linear Matrix Inequality (LMI) constraints. If the constraint set is found to be feasible we search for a minimal representation; otherwise, we search for an irreducible infeasible system. The work is based on the solution of a set covering problem wh ..."
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We present a constraint analysis methodology for Linear Matrix Inequality (LMI) constraints. If the constraint set is found to be feasible we search for a minimal representation; otherwise, we search for an irreducible infeasible system. The work is based on the solution of a set covering problem where each row corresponds to a sample point and is determined by constraint satisfaction at the sampled point. Thus, an implementation requires a method to collect points in the ambient space and a constraint oracle. Much of this paper will be devoted to the development of a hit and run sampling methodology. Test results confirm that our approach not only provides information required for constraint analysis, but will also, if the feasible region has a nonvoid interior, with probability one, find a feasible point. Key words: linear matrix inequalities; positive semidefinite programming; feasibility; redundancy; irreducible infeasible sets 1.