Convex Cone Research Articles

A THEOREM OF TYPE “PARTITION OF UNITY” WITH

In the first part of the paper, we present a theorem of type “partition of unity”, and three of its consequences, i.e., for algebras, linear subspaces and convex cones. In the second part, some theorems of localizability (or density) in a weighted space are presented. We mention that the weighted spaces are classes of continuous scalar functions on a locally compact space (for example, the space of function with compact support, the space of bounded functions, the space of functions vanishing at infinity, the space of functions rapidly decreasing at infinity).

Open-loop equilibrium strategy for mean-variance Portfolio selection with investment constraints in a non-Markovian regime-switching jump-diffusion model

<p style='text-indent:20px;'>This paper is devoted to study the open-loop equilibrium strategy for a mean-variance portfolio problem with investment constraints in a non-Markovian regime-switching jump-diffusion model. Specially, the investment strategies are constrained in a closed convex cone and all coefficients in the model are stochastic processes adapted to the filtration generated by a Markov chain. First, we provide a necessary and sufficient condition for an equilibrium strategy, which involves a system of forward and backward stochastic differential equations (FBSDEs, for short). Second, by solving these FBSDEs, we obtain a feedback representation of the equilibrium strategy. Third, we prove a theorem ensuring the almost everywhere uniqueness of the equilibrium solution. Finally, the results are applied to solve an example of the Markovian regime-switching model.</p>

Конусное обобщение теоремы Банаха и накрывание вдоль кривых

This work is devoted to the study of the covering property of linear and nonlinear mappings of Banach spaces. We consider linear continuous operators acting from one Banach space to another. For a given operator, it is shown that for any point y_0 from the relative interior of the image of a given convex closed cone there exists a conical neighborhood of y_0, with respect to which the given operator has the covering property at zero with a covering constant depending on the point y0: We provide an example showing that for a linear continuous operator the covering property with respect to the image of a given cone at zero may fail, i. e. the statement of Banach’s theorem on an open mapping may not hold for restrictions of linear continuous operators to closed convex cones. We obtain a corollary of the obtained theorem for the case when the target space is finite-dimensional. Moreover, nonlinear twice differentiable mappings of Banach spaces are considered. For them, conditions for local covering along a certain curve with respect to a given cone are presented. The corresponding sufficient conditions are formulated in terms of 2 -regular directions. They remain meaningful even in the case of degeneracy of the first derivative of the mapping under consideration at a given point.

Proper single splittings over proper cones of rectangular matrices

Proper single splittings over proper cones of rectangular matrices

Annihilating Entanglement Between Cones

Every multipartite entangled quantum state becomes fully separable after an entanglement breaking quantum channel acted locally on each of its subsystems. Whether there are other quantum channels with this property has been an open problem with important implications for entanglement theory (e.g., for the distillation problem and the PPT squared conjecture). We cast this problem in the general setting of proper convex cones in finite-dimensional vector spaces. The max-entanglement annihilating maps transform the k-fold maximal tensor product of a cone {textsf{C}}_1 into the k-fold minimal tensor product of a cone {textsf{C}}_2, and the pair ({textsf{C}}_1,{textsf{C}}_2) is called resilient if all max-entanglement annihilating maps are entanglement breaking. Our main result is that ({textsf{C}}_1,{textsf{C}}_2) is resilient if either {textsf{C}}_1 or {textsf{C}}_2 is a Lorentz cone. Our proof exploits the symmetries of the Lorentz cones and applies two constructions resembling protocols for entanglement distillation: As a warm-up, we use the multiplication tensors of real composition algebras to construct a finite family of generalized distillation protocols for Lorentz cones, containing the distillation protocol for entangled qubit states by Bennett et al. (Phys Rev Lett 76(5):722, 1996) as a special case. Then, we construct an infinite family of protocols using solutions to the Hurwitz matrix equations. After proving these results, we focus on maps between cones of positive semidefinite matrices, where we derive necessary conditions for max-entanglement annihilation similar to the reduction criterion in entanglement distillation. Finally, we apply results from the theory of Banach space tensor norms to show that the Lorentz cones are the only cones with a symmetric base for which a certain stronger version of the resilience property is satisfied.

A characterization for the self-duality of proper cones

A characterization for the self-duality of proper cones

SMEFTs living on the edge: determining the UV theories from positivity and extremality

We study the “inverse problem” in the context of the Standard Model Effective Field Theory (SMEFT): how and to what extend can one reconstruct the UV theory, given the measured values of the operator coefficients in the IR? The main obstacle of this problem is the degeneracies in the space of coefficients: a given SMEFT truncated at a finite dimension can be mapped to infinitely many UV theories. We discuss these degeneracies at the dimension-8 level, and show that positivity bounds play a crucial role in the inverse problem. In particular, the degeneracies either vanish or become significantly limited for SMEFTs that live on or close to the positivity bounds. The UV particles of these SMEFTs, and their properties such as spin, charge, other quantum numbers, and interactions with the SM particles, can often be uniquely determined, assuming dimension-8 coefficients are measured. The allowed region for SMEFTs, which forms a convex cone, can be systematically constructed by enumerating its generators. We show that a geometric notion, extremality, conveniently connects the positivity problem with the inverse problem. We discuss the implications of a SMEFT living on an extremal ray, on a k-face, and on the vertex of the positive cone. We also show that the information of the dimension-8 coefficients can be used to set exclusion limits on all individual UV states that interact with the SM, independent of specific model assumptions. Our results indicate that the dimension-8 operators encode much more information about the UV than one would naively expect, which can be used to reverse engineer the UV physics from the SMEFT.

Gradient boosting for convex cone predict and optimize problems

Prediction models are traditionally optimized independently from decision-based optimization. Conversely, a ‘smart predict then optimize’ (SPO) framework optimizes prediction models to minimize downstream decision regret. In this paper we present dboost, the first general purpose implementation of smart gradient boosting for ‘predict, then optimize’ problems. The framework supports convex quadratic cone programming and gradient boosting is performed by implicit differentiation of a custom fixed-point mapping. Experiments comparing with state-of-the-art SPO methods show that dboost can further reduce out-of-sample decision regret.

Radial symmetry and partially overdetermined problems in a convex cone

AbstractWe obtain the radial symmetry of the solution to a partially overdetermined boundary value problem in a convex cone in space forms by using the maximum principle for a suitable subharmonic function P and integral identities. In dimension 2, we prove Serrin‐type results for partially overdetermined problems outside a convex cone. Furthermore, we obtain a Rellich identity for an eigenvalue problem with mixed boundary conditions in a cone.

Conic Optimization with Spectral Functions on Euclidean Jordan Algebras

Spectral functions on Euclidean Jordan algebras arise frequently in convex optimization models. Despite the success of primal-dual conic interior point solvers, there has been little work on enabling direct support for spectral cones, that is, proper nonsymmetric cones defined from epigraphs and perspectives of spectral functions. We propose simple logarithmically homogeneous barriers for spectral cones and we derive efficient, numerically stable procedures for evaluating barrier oracles such as inverse Hessian operators. For two useful classes of spectral cones—the root-determinant cones and the matrix monotone derivative cones—we show that the barriers are self-concordant, with nearly optimal parameters. We implement these cones and oracles in our open-source solver Hypatia, and we write simple, natural formulations for four applied problems. Our computational benchmarks demonstrate that Hypatia often solves the natural formulations more efficiently than advanced solvers such as MOSEK 9 solve equivalent extended formulations written using only the cones these solvers support.Funding: This work was supported by Office of Naval Research [Grant N00014-18-1-2079] and the National Science Foundation [Grant OAC-1835443].

Monotonicity results for non-autonomous dynamical systems: The case of a general convex cone

Let K K be a closed convex cone in the state space R n \mathbb {R}^n . This note characterizes the K K -monotonicity of a non-autonomous dynamical system x ˙ ( t ) = f ( t , x ( t ) ) \dot x (t) = f(t,x(t)) governed by a locally Lipschitz velocity field. We deviate from the classical literature in two important ways. Firstly, the velocity field f f is not required to be differentiable with respect to the state variables. And, secondly, the closed convex cone K K is allowed to be absolutely general. In particular, we impose neither pointedness, nor solidity.

A note on the real support of Silva–Hasumi–Morimoto ultrahyperfunctions

Recently, we have studied the relation between the singular spectrum of a class of tempered ultrahyperfunctions corresponding to proper convex cones and their expressions as boundary values of holomorphic functions. In this note, we added some more results on the singular spectrum of the tempered ultrahyperfunctions. More specifically, the “microlocalization” of a Silva–Hasumi–Morimoto ultrahyperfunction corresponding to an open proper convex cone is addressed. Following Nishiwada’s approach to the analytic wave front set, we introduce the real singular spectrum of such an ultrahyperfunction and prove that it canonically projects onto Silva’s real support.

Hypersurfaces of constant Gauss–Kronecker curvature with Li-normalization in affine space

For convex hypersurfaces in the affine space \({\mathbb {A}}^{n+1}\) (\(n\ge 2\)), A.-M. Li introduced the notion of \(\alpha \)-normal field as a generalization of the affine normal field. By studying a Monge–Ampère equation with gradient blowup boundary condition, we show that regular domains in \({\mathbb {A}}^{n+1}\), defined with respect to a proper convex cone and satisfying some regularity assumption if \(n\ge 3\), are foliated by complete convex hypersurfaces with constant Gauss–Kronecker curvature relative to the Li-normalization. When \(n=2\), a key feature is that no regularity assumption is required, and the result extends our recent work about the \(\alpha =1\) case.

Evaluating automated endmember extraction for classifying hyperspectral data and deriving spectral parameters for monitoring forest vegetation health.

The hyperspectral remote sensing datasets possess high capability in differentiating the spectrally similar features, and thus, they are immensely important in various forestry activities, especially vegetation classifications. But extracting endmembers for data training is a challenging task. The present study is focused on the use of automated endmember extraction technique for deriving endmembers during the unavailability of ground spectra. We used the Sequential Maximum Angle Convex Cone (SMACC) method on EO-1 Hyperion data for endmember extraction in the Barkot forest range of Dehradun district, Uttarakhand which were used for classification of the study area using support vector machine (SVM). Further, we estimated the vegetation health of the region by assigning the threshold weights for various derived environmental variables such as NDVI (Normalised Difference Vegetation Index), CRI (Carotenoid Reflectance Index), Anthocyanin Reflectance Index (ARI), Modified Simple Ratio (MSR), Modified Chlorophyll Absorption Ratio Index (MCARI) and WBI (Water Band Index). Then, to further validate the health of the forest types, we correlated it with the Land Surface Temperature (LST) from LANDSAT 5 ETM + data. The results showed a high classification accuracy of 89.13%. The healthy vegetation area coverage of the area was about 78.6% with most healthy class as Tectona grandis and Shorea robusta and its correlation with LST showed lower temperature range in healthy vegetation areas and vice versa. The study was useful in determining the superiority of SMACC automated endmember extraction and estimating the vegetation health.

What Should a Firm Know? Protecting Consumers’ Privacy Rents

A monopolistic firm observes a signal about the state of the world and then makes a take-it-or-leave-it offer to an uninformed consumer who has recourse to some outside option. We provide a geometric characterization of the firm’s information structure that maximizes the consumer’s surplus: the optimal regime partitions the space of payoff states into polyhedral cones with disjoint interiors. We interpret our results in terms of the maximization of the consumer’s “privacy rent.” We illustrate and motivate our approach through the example of the regulation of the privacy of medical information in monopolistic health insurance markets. (JEL D21, D42, D83, G22, I13)

Remarks on nef and movable cones of hypersurfaces in Mori dream spaces

We investigate nef and movable cones of hypersurfaces in Mori dream spaces. The first result is: Let Z be a smooth Mori dream space of dimension at least four whose extremal contractions are of fiber type of relative dimension at least two and let X be a smooth ample divisor in Z, then X is a Mori dream space as well. The second result is: Let Z be a Fano manifold of dimension at least four whose extremal contractions are of fiber type and let X be a smooth anti-canonical hypersurface in Z, which is a smooth Calabi–Yau variety, then the unique minimal model of X up to isomorphism is X itself, and moreover, the movable cone conjecture holds for X, namely, there exists a rational polyhedral cone which is a fundamental domain for the action of birational automorphisms on the effective movable cone of X. The third result is: Let P:=Pn×⋯×Pn be the N-fold self-product of the n-dimensional projective space. Let X be a general complete intersection of n+1 hypersurfaces of multidegree (1,…,1) in P with dim⁡X≥3. Then X has only finitely many minimal models up to isomorphism, and moreover, the movable cone conjecture holds for X.

System Monotonicity and Subspace Tracking: A Geometric Perspective of the Frisch–Shapiro Scheme

The Shapiro scheme, together with the closely related Frisch–Kalman scheme, has been an important approach to system identification and statistical analysis. A longstanding result on this scheme, known as the Shapiro theorem, is both informative and significant. This article imparts a geometric understanding to the Shapiro theorem and generalizes it to the asymmetric setting using the notion of cone-invariance. In particular, we establish the equivalence between two important properties of a real-valued square matrix—irreducible orthant-invariance and simplicity of its dominant eigenvalue under arbitrary diagonal perturbations. The result can be regarded as a converse Perron–Frobenius theorem. Furthermore, we investigate two applications of the proposed result in systems and control, namely, characterization of irreducibly orthant-monotone nonlinear systems and subspace tracking via decentralized control. We also extend the established result to accommodating polyhedral cones and obtain several insights.

Implementing a fast unbounded quantum fanout gate using power-law interactions

The standard circuit model for quantum computation presumes the ability to directly perform gates between arbitrary pairs of qubits, which is unlikely to be practical for large-scale experiments. Power-law interactions with strength decaying as $1/r^\alpha$ in the distance $r$ provide an experimentally realizable resource for information processing, whilst still retaining long-range connectivity. We leverage the power of these interactions to implement a fast quantum fanout gate with an arbitrary number of targets. Our implementation allows the quantum Fourier transform (QFT) and Shor's algorithm to be performed on a $D$-dimensional lattice in time logarithmic in the number of qubits for interactions with $\alpha \le D$. As a corollary, we show that power-law systems with $\alpha \le D$ are difficult to simulate classically even for short times, under a standard assumption that factoring is classically intractable. Complementarily, we develop a new technique to give a general lower bound, linear in the size of the system, on the time required to implement the QFT and the fanout gate in systems that are constrained by a linear light cone. This allows us to prove an asymptotically tighter lower bound for long-range systems than is possible with previously available techniques.

The quasi-static plasmonic problem for polyhedra

We characterize the essential spectrum of the plasmonic problem for polyhedra in {mathbb {R}}^3. The description is particularly simple for convex polyhedra and permittivities epsilon < - 1. The plasmonic problem is interpreted as a spectral problem through a boundary integral operator, the direct value of the double layer potential, also known as the Neumann–Poincaré operator. We therefore study the spectral structure of the double layer potential for polyhedral cones and polyhedra.

A class of inverse curvature flows and 𝐿^{𝑝} dual Christoffel-Minkowski problem

In this paper, we consider a large class of expanding flows of closed, smooth, star-shaped hypersurface in Euclidean space R n + 1 \mathbb {R}^{n+1} with speed ψ u α ρ δ f − β \psi u^\alpha \rho ^\delta f^{-\beta } , where ψ \psi is a smooth positive function on unit sphere, u u is the support function of the hypersurface, ρ \rho is the radial function, f f is a smooth, symmetric, homogenous of degree one, positive function of the principal curvatures of the hypersurface on a convex cone. When ψ = 1 \psi =1 , we prove that the flow exists for all time and converges to infinity if α + δ + β ⩽ 1 \alpha +\delta +\beta \leqslant 1 , and α ⩽ 0 &gt; β \alpha \leqslant 0&gt;\beta , while in case α + δ + β &gt; 1 \alpha +\delta +\beta &gt;1 , α , δ ⩽ 0 &gt; β \alpha ,\delta \leqslant 0&gt;\beta , the flow blows up in finite time, and where we assume the initial hypersurface to be strictly convex. In both cases the properly rescaled flows converge to a sphere centered at the origin. In particular, the results of Gerhardt [J. Differential Geom. 32 (1990), pp. 299–314; Calc. Var. Partial Differential Equations 49 (2014), pp. 471–489] and Urbas [Math. Z. 205 (1990), pp. 355–372] can be recovered by putting α = δ = 0 \alpha =\delta =0 . Our previous works [Proc. Amer. Math. Soc. 148 (2020), pp. 5331–5341; J. Funct. Anal. 282 (2022), p. 38] and Hu, Mao, Tu and Wu [J. Korean Math. Soc. 57 (2020), pp. 1299–1322] can be recovered by putting δ = 0 \delta =0 and α = 0 \alpha =0 respectively. By the convergence of these flows, we can give a new proof of uniqueness theorems for solutions to L p L^p -Minkowski problem and L p L^p -Christoffel-Minkowski problem with constant prescribed data. Similarly, we consider the L p L^p dual Christoffel-Minkowski problem and prove a uniqueness theorem for solutions to L p L^p dual Minkowski problem and L p L^p dual Christoffel-Minkowski problem with constant prescribed data. At last, we focus on the long time existence and convergence of a class of anisotropic flows (i.e. for general function ψ \psi ). The final result not only gives a new proof of many previously known solutions to L p L^p dual Minkowski problem, L p L^p -Christoffel-Minkowski problem, etc. by such anisotropic flows, but also provides solutions to L p L^p dual Christoffel-Minkowski problem with some conditions.