Vector Optimization w.r.t. Relatively Solid Convex Cones in Real Linear Spaces

In vector optimization, it is of increasing interest to study problems where the image space (a real linear space) is preordered by a not necessarily solid (and not necessarily pointed) convex cone. It is well-known that there are many examples where the ordering cone of the image space has an empty (topological/algebraic) interior, for instance in optimal control, approximation theory, duality theory. Our aim is to consider Pareto-type solution concepts for such vector optimization problems based on the intrinsic core notion (a well-known generalized interiority notion). We propose a new Henig-type proper efficiency concept based on generalized dilating cones which are relatively solid (i.e., their intrinsic cores are nonempty). Using linear functionals from the dual cone of the ordering cone, we are able to characterize the sets of (weakly, properly) efficient solutions under certain generalized convexity assumptions. Toward this end, we employ separation theorems that are working in the considered setting.

The dual cone of sums of non-negative circuit polynomials

For a non-empty, finite subset A ⊆ N 0 n $\mathcal{A} \subseteq \mathbb{N}_0^n$ let C sonc(𝒜) ∈ ℝ[x 1, . . . , xn ] be the cone of sums of non-negative circuit polynomials with support 𝒜. We derive a representation of the dual cone (C sonc(𝒜))∗ and deduce an optimality criterion for sums of non-negative circuit polynomials in polynomial optimization.

Geometry Calibration of a Modular Stereo Cone-Beam X-ray CT System

Compared to single source systems, stereo X-ray CT systems allow acquiring projection data within a reduced amount of time, for an extended field-of-view, or for dual X-ray energies. To exploit the benefit of a dual X-ray system, its acquisition geometry needs to be calibrated. Unfortunately, in modular stereo X-ray CT setups , geometry misalignment occurs each time the setup is changed, which calls for an efficient calibration procedure. Although many studies have been dealing with geometry calibration of an X-ray CT system, little research targets the calibration of a dual cone-beam X-ray CT system. In this work, we present a phantom-based calibration procedure to accurately estimate the geometry of a stereo cone-beam X-ray CT system. With simulated as well as real experiments, it is shown that the calibration procedure can be used to accurately estimate the geometry of a modular stereo X-ray CT system thereby reducing the misalignment artifacts in the reconstruction volumes.

Orthogonality in smooth countably normed spaces

We generalize the concepts of normalized duality mapping, J-orthogonality and Birkhoff orthogonality from normed spaces to smooth countably normed spaces. We give some basic properties of J-orthogonality in smooth countably normed spaces and show a relation between J-orthogonality and metric projection on smooth uniformly convex complete countably normed spaces. Moreover, we define the J-dual cone and J-orthogonal complement on a nonempty subset S of a smooth countably normed space and prove some basic results about the J-dual cone and the J-orthogonal complement of S.

Extreme Ray Feasibility Cuts for Unit Commitment with Uncertainty

The unit commitment problem with uncertainty is considered one of the most challenging power system scheduling problems. Different stochastic models have been proposed to solve the problem, but such approaches have yet to be applied in industry practice because of computational challenges. In practice, the problem is formulated as a deterministic model with reserve requirements to hedge against uncertainty. However, simply requiring a certain level of reserves cannot ensure power system reliability as the procured reserves may be nondispatchable because of transmission limitations. In this paper, we derive a set of feasibility cuts (constraints) for managing the unit commitment problem with uncertainty. These cuts eliminate unreliable scheduling solutions and reallocate reserves in the power system; they are induced by the extreme rays of a polyhedral dual cone. This paper shows that, with the proposed reformulation, the extreme rays of the dual cone can be characterized by combinatorial selections of transmission lines (arcs) and buses (nodes) of the power system. As a result, the cuts can then be characterized using engineering insights. The unit commitment problem with uncertainty is formulated as a deterministic model with the identified extreme ray feasibility cuts. Test results show that, with the proposed extreme ray feasibility cuts, the problem can be solved more efficiently, and the resulting scheduling decision is also more reliable.

DICOM compression and decompression method using double cone

The increase of information in the medical environment caused by digital imaging settings is notable. The search and use of these technological tools aimed at medicine require a greater availability of storage, generating increasing costs. In medicine, together with information technology, there is a format of images used in exams, diagnostics, tomography, among others. This format, entitled DICOM, was created in order to standardize uses in medical devices for exam answers. An open question is the compression of DICOM data, in order to maintain quality, maintaining high rates of compression. This presents a new method for compressing and decompressing DICOM data using a dual cone bijector function and a video codec, called DC (Double Cone). This work offers 3 changes to the DC method (DC1, DC2 and DC3). The results obtained with a new technique show that the compression, although with loss, has a similarity index very close to the original image (SSIM = 0.99), and an accuracy ratio equal to 69.51, in the better case. The better performing version was the DC2.

Lagrange multiplier characterizations of constrained best approximation with nonsmooth nonconvex constraints

In this paper, we consider the constraint set K := {x ? Rn : gj(x)? 0,? j = 1,2,...,m} of inequalities with nonsmooth nonconvex constraint functions gj : Rn ? R (j = 1,2,...,m).We show that under Abadie?s constraint qualification the ?perturbation property? of the best approximation to any x in Rn from a convex set ?K := C ? K is characterized by the strong conical hull intersection property (strong CHIP) of C and K, where C is an arbitrary non-empty closed convex subset of Rn: By using the idea of tangential subdifferential and a non-smooth version of Abadie?s constraint qualification, we do this by first proving a dual cone characterization of the constraint set K. Moreover, we present sufficient conditions for which the strong CHIP property holds. In particular, when the set ?K is closed and convex, we show that the Lagrange multiplier characterizations of constrained best approximation holds under a non-smooth version of Abadie?s constraint qualification. The obtained results extend many corresponding results in the context of constrained best approximation. Several examples are provided to clarify the results.

Cone-constrained rational eigenvalue problems

This work deals with the eigenvalue analysis of a rational matrix-valued function subject to complementarity constraints induced by a polyhedral cone $K$. The eigenvalue problem under consideration has the general structure \[ \left(\sum_{k=0}^d \lambda^k A_k + \sum_{k =1}^m \frac{p_k(\lambda)}{q_k(\lambda)} \,B_k\right) x = y , \quad K\ni x \perp y\in K^\ast, \] where $K^\ast$ denotes the dual cone of $K$. The unconstrained version of this problem has been discussed in [Y.F. Su and Z.J. Bai. Solving rational eigenvalue problems via linearization. \emph{SIAM J. Matrix Anal. Appl.}, 32:201--216, 2011.] with special emphasis on the implementation of linearization-based methods. The cone-constrained case can be handled by combining Su and Bai's linearization approach and the so-called facial reduction technique. In essence, this technique consists in solving one unconstrained rational eigenvalue problem for each face of the polyhedral cone $K$.