242,248 research outputs found
Necessary and sufficient condition on global optimality without convexity and second order differentiability
The main goal of this paper is to give a necessary and sufficient condition
of global optimality for unconstrained optimization problems, when the objective
function is not necessarily convex. We use Gâteaux differentiability of the objective
function and its bidual (the latter is known from convex analysis)
Behavioral analysis of anisotropic diffusion in image processing
©1996 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or distribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE. This material is presented to ensure timely dissemination of scholarly and technical work. Copyright and all rights therein are retained by authors or by other copyright holders. All persons copying this information are expected to adhere to the terms and constraints invoked by each author's copyright. In most cases, these works may not be reposted without the explicit permission of the copyright holder.DOI: 10.1109/83.541424In this paper, we analyze the behavior of the anisotropic diffusion model of Perona and Malik (1990). The main idea is to express the anisotropic diffusion equation as coming from a certain optimization problem, so its behavior can be analyzed based on the shape of the corresponding energy surface. We show that anisotropic diffusion is the steepest descent method for solving an energy minimization problem. It is demonstrated that an anisotropic diffusion is well posed when there exists a unique global minimum for the energy functional and that the ill posedness of a certain anisotropic diffusion is caused by the fact that its energy functional has an infinite number of global minima that are dense in the image space. We give a sufficient condition for an anisotropic diffusion to be well posed and a sufficient and necessary condition for it to be ill posed due to the dense global minima. The mechanism of smoothing and edge enhancement of anisotropic diffusion is illustrated through a particular orthogonal decomposition of the diffusion operator into two parts: one that diffuses tangentially to the edges and therefore acts as an anisotropic smoothing operator, and the other that flows normally to the edges and thus acts as an enhancement operator
Least-squares approximation of an improper by a proper correlation matrix using a semi-infinite convex program
An algorithm is presented for the best least-squares fitting correlation matrix approximating a given missing value or improper correlation matrix. The proposed algorithm is based on a solution for C. I. Mosier's oblique Procrustes rotation problem offered by J. M. F. ten Berge and K. Nevels (1977). It is shown that the minimization problem belongs to a certain class of convex programs in optimization theory. A necessary and sufficient condition for a solution to yield the unique global minimum of the least-squares function is derived from a theorem by A. Shapiro (1985). A computer program was implemented to yield the solution of the minimization problem with the proposed algorithm. This empirical verification of the condition indicates that the occurrence of non-optimal solutions with the proposed algorithm is very unlikely
Saving phase: Injectivity and stability for phase retrieval
Recent advances in convex optimization have led to new strides in the phase
retrieval problem over finite-dimensional vector spaces. However, certain
fundamental questions remain: What sorts of measurement vectors uniquely
determine every signal up to a global phase factor, and how many are needed to
do so? Furthermore, which measurement ensembles lend stability? This paper
presents several results that address each of these questions. We begin by
characterizing injectivity, and we identify that the complement property is
indeed a necessary condition in the complex case. We then pose a conjecture
that 4M-4 generic measurement vectors are both necessary and sufficient for
injectivity in M dimensions, and we prove this conjecture in the special cases
where M=2,3. Next, we shift our attention to stability, both in the worst and
average cases. Here, we characterize worst-case stability in the real case by
introducing a numerical version of the complement property. This new property
bears some resemblance to the restricted isometry property of compressed
sensing and can be used to derive a sharp lower Lipschitz bound on the
intensity measurement mapping. Localized frames are shown to lack this property
(suggesting instability), whereas Gaussian random measurements are shown to
satisfy this property with high probability. We conclude by presenting results
that use a stochastic noise model in both the real and complex cases, and we
leverage Cramer-Rao lower bounds to identify stability with stronger versions
of the injectivity characterizations.Comment: 22 page
Control of production-distribution systems under discrete disturbances and control actions.
This paper deals with the robust control and optimization of production-distribution systems. The model used in our problem formulation is a general network flow model that describes production, logistics, and transportation
applications. The novelty in our formulation is in the discrete nature of the control and disturbance inputs. We highlight three main contributions: First, we derive a necessary and sufficient condition for the existence of robustly control invariant hyperboxes. Second, we show that a stricter version of the same condition is sufficient for global convergence to an invariant set. Third, for the scalar case, we show that these results parallel existing results in the setting where the control actions and disturbances are analog. We conclude with two simple illustrative examples
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