The basis pursuit problem seeks a minimum one-norm solution of an underdetermined least-squares problem. Basis pursuit denoise (BPDN) fits the least-squares problem only approximately, and a single parameter determine...
详细信息
The basis pursuit problem seeks a minimum one-norm solution of an underdetermined least-squares problem. Basis pursuit denoise (BPDN) fits the least-squares problem only approximately, and a single parameter determines a curve that traces the optimal trade-off between the least-squares fit and the one-norm of the solution. We prove that this curve is convex and continuously differentiable over all points of interest, and show that it gives an explicit relationship to two other optimization problems closely related to BPDN. We describe a root-finding algorithm for finding arbitrary points on this curve;the algorithm is suitable for problems that are large scale and for those that are in the complex domain. At each iteration, a spectral gradient-projection method approximately minimizes a least-squares problem with an explicit one-norm constraint. Only matrix-vector operations are required. The primal-dual solution of this problem gives function and derivative information needed for the root-finding method. Numerical experiments on a comprehensive set of test problems demonstrate that the method scales well to large problems.
With the objective of generating "shape-preserving" smooth interpolating curves that represent data with abrupt changes in magnitude and/or knot spacing, we study a class of first-derivative-based C-1-smooth...
详细信息
With the objective of generating "shape-preserving" smooth interpolating curves that represent data with abrupt changes in magnitude and/or knot spacing, we study a class of first-derivative-based C-1-smooth univariate cubic L-1 splines. An L-1 spline minimizes the L-1 norm of the difference between the first-order derivative of the spline and the local divided difference of the data. Calculating the coefficients of an L-1 spline is a nonsmooth non-linear convex program. Via Fenchel's conjugate transformation, the geometric dual program is a smooth convex program with a linear objective function and convex cubic constraints. The dual-to-primal transformation is accomplished by solving a linear program.
For positive time between proposals, stationary SPE (physical) allocations in alternating offers bargaining correspond to an asymmetric Nash bargaining solution. By the Maximum Theorem, convergence when time between p...
详细信息
For positive time between proposals, stationary SPE (physical) allocations in alternating offers bargaining correspond to an asymmetric Nash bargaining solution. By the Maximum Theorem, convergence when time between proposals vanishes is immediate. Theoretical results are extended, numerical implementation is straightforward. (C) 2007 Elsevier B.V All rights reserved.
This paper studies a difficult and fundamental problem that arises throughout electrical engineering, applied mathematics, and statistics. Suppose that one forms a short linear combination of elementary signals drawn ...
详细信息
This paper studies a difficult and fundamental problem that arises throughout electrical engineering, applied mathematics, and statistics. Suppose that one forms a short linear combination of elementary signals drawn from a large, fixed collection. Given an observation of the linear combination that has been contaminated with additive noise, the goal is to identify which elementary signals participated and to approximate their coefficients. Although many algorithms have been proposed, there is little theory which guarantees that these algorithms can accurately and efficiently solve the problem. This paper studies a method called convex relaxation, which attempts to recover the ideal sparse signal by solving a convex program. This approach is powerful because the optimization can be completed in polynomial time with standard scientific software. The paper provides general conditions which ensure that convex relaxation succeeds. As evidence of the broad impact of these results, the paper describes how convex relaxation can be used for several concrete signal recovery problems. It also describes applications to channel coding, linear regression, and numerical analysis.
We consider convex constrained optimization problems, and we enhance the classical Fritz John optimality conditions to assert the existence of multipliers with special sensitivity properties. In particular, we prove t...
详细信息
We consider convex constrained optimization problems, and we enhance the classical Fritz John optimality conditions to assert the existence of multipliers with special sensitivity properties. In particular, we prove the existence of Fritz John multipliers that are informative in the sense that they identify constraints whose relaxation, at rates proportional to the multipliers, strictly improves the primal optimal value. Moreover, we show that if the set of geometric multipliers is nonempty, then the minimum-norm vector of this set is informative and defines the optimal rate of cost improvement per unit constraint violation. Our assumptions are very general and, in particular, allow for the presence of a duality gap and the nonexistence of optimal solutions. In particular, for the case where there is a duality gap, we establish enhanced Fritz John conditions involving the dual optimal value and dual optimal solutions.
This study proposes a novel method to solve nonlinear fractional programming (NFP) problems occurring frequently in engineering design and management. Fractional terms composed of signomial functions are first decompo...
详细信息
This study proposes a novel method to solve nonlinear fractional programming (NFP) problems occurring frequently in engineering design and management. Fractional terms composed of signomial functions are first decomposed into convex and concave terms by convexification strategies. Then the piecewise linearization technique is used to approximate the concave terms. The NFP program is then converted into a convex program. A global optimum of the fractional program can finally be found within a tolerable error. When compared with most of the current fractional programming methods, which can only treat a problem with linear functions or a single quotient term, the proposed method can solve a more general fractional programming program with nonlinear functions and multiple quotient terms. Numerical examples are presented to demonstrate the usefulness of the proposed method.
In this paper, we consider an important fuzzy version of the well known smallest covering circle problem which is also called the Euclidean 1-center problem. Here we are given a set of points in the plane whose locati...
详细信息
In this paper, we consider an important fuzzy version of the well known smallest covering circle problem which is also called the Euclidean 1-center problem. Here we are given a set of points in the plane whose locations are crisp. However, the requirement for coverage of each point is fuzzy as represented by its grade of membership. As such we qualify the points as fuzzy. In the above framework, we wish to find a fuzzy disk with minimum fuzzy area for covering the given fuzzy points. After developing a general model, certain special cases are investigated in detail. Polynomial algorithms are presented for the special cases. (C) 2003 Elsevier B.V. All rights reserved.
Second-order optimality conditions are studied for the constrained optimization problem where the objective function and the constraints are compositions of convex functions and twice strictly differentiable functions...
详细信息
Second-order optimality conditions are studied for the constrained optimization problem where the objective function and the constraints are compositions of convex functions and twice strictly differentiable functions. A second-order sufficient condition of a global minimizer is obtained by introducing a generalized representation condition. Second-order minimizer characterizations for a convex program and a linear fractional program are derived using the generalized representation condition.
Second-order optimality conditions are studied for the constrained optimization problem where the objective function and the constraints are compositions of convex functions and twice strictly differentiable functions...
详细信息
Second-order optimality conditions are studied for the constrained optimization problem where the objective function and the constraints are compositions of convex functions and twice strictly differentiable functions. A second-order sufficient condition of a global minimizer is obtained by introducing a generalized representation condition. Second-order minimizer characterizations for a convex program and a linear fractional program are derived using the generalized representation condition.
Dynamic stochastic programs are prototypical for optimization problems with an inherent tree structure inducing characteristic sparsity patterns in the KKT systems of interior methods. We propose an integrated modelin...
详细信息
Dynamic stochastic programs are prototypical for optimization problems with an inherent tree structure inducing characteristic sparsity patterns in the KKT systems of interior methods. We propose an integrated modeling and solution approach for such tree-sparse programs. Three closely related natural formulations are theoretically analyzed from a control-theoretic perspective and compared to each other. Associated KKT system solution algorithms with linear complexity are developed and comparisons to other interior approaches and related problem formulations are discussed.
暂无评论