The paper studies quasi-X-convex functions and their applications in optimization.
problem Optimization problems with quasi-X-convex functions. method Definition and study of X-convex, quasi-X-convex, and related functions. result Applications of quasi-X-convex functions in optimization problems. Classifies ancient convex curves in convex domains.
problem Ancient convex curve shortening flows on convex domains.
method Classification of convex ancient solutions.
result Ancient convex curves in convex domains classified.
Strict convexity proven for certain self-expanders in high dimensions.
problem Convexity of self-expanders in mean curvature flow.
method Investigation of convexity properties for asymptotically conical self-expanders.
result Strict convexity proven for n-dimensional self-expanders. Convex optimization is a vibrant and successful area due to the existence of a variety of efficient algorithms that leverage the rich structure provided by convexity. Convexity of a smooth set or a function in a Euclidean space is defined by how it interacts with the standard differential structure in this space -- the…
The paper proves convexity of certain solitons and expanders in high dimensions.
problem Proving convexity of specific solitons and expanders in Rn+1. method Inspired by Spruck-Xiao and Derdziński, the paper uses geometric analysis to prove convexity.
result The paper proves the convexity of complete 2-convex translating and expanding solitons and expanders in Rn+1 for n≥3. Introduces GG-convex risk measures and derives their dual representations.
problem Defining and studying GG-convex risk measures.
method Introduces GG-convex conjugate, derives dual representations, and studies Orlicz risk measures.
result Derives a general dual representation for GG-convex risk measures.
New geometric proof of convex function differentiability and approximation.
problem Second-order differentiability of convex functions and their approximations.
method Elementary geometric approach to prove classical and recent results.
result New proofs of Lusin approximation of convex functions and bodies by C1,1 functions. Optimal risk sharing without convex preferences using aggregate convexity.
problem Risk sharing among non-convex preferences.
method Aggregate convexity principles and Lyapunov convexity, combined with approximation arguments for law invariant risk measures.
result Derivation of a computationally tractable formula for the conjugate of the value function.
Convex clustering can only learn convex clusters, with significant gaps between clusters.
problem Understanding the limitations and capabilities of convex clustering.
method Analyzing convex clustering solutions, proving properties, and characterizing clusters.
result Convex clustering can only learn convex clusters with significant gaps between clusters.
Extends DCP framework to Hadamard manifolds for geodesically convex functions.
problem Verifying convexity in nonlinear programs on Hadamard manifolds.
method Introduces Disciplined Geodesically Convex Programming (DGCP) framework, defining compositions and transformations for geodesically convex functions.
result Allows verification of geodesic convexity for a broader range of functions, including statistical estimators and matrix-valued optimization.
Proves convexity of certain hypersurfaces with negative λ.
problem Understanding convexity of hypersurfaces with specific λ values.
method Analyzes mean convex hypersurfaces and proves convexity for λ ≤ 0.
result Closed n-dimensional mean convex λ-hypersurfaces are convex if λ≤0. New convexity concept applied to sphere yields quermassintegral inequalities.
problem Proving quermassintegral inequalities for horo-convex hypersurfaces on the sphere.
method Smooth convergence of Guan/Li flow for inverse type applied to horo-convex hypersurfaces.
result Full set of quermassintegral inequalities for horo-convex hypersurfaces proved.
In this paper, we study compact convex Lefschetz fibrations on compact convex symplectic manifolds (i.e., Liouville domains) of dimension 2n+2 which are introduced by Seidel and later also studied by McLean. By a result of Akbulut-Arikan, the open book on ∂W, which we call \emph{convex open book}, induced b…
Every convex set in a generic Riemannian manifold has peculiar properties.
problem Characterizing convex sets in Riemannian manifolds.
method Analyzing geodesics and hypersurfaces in Riemannian manifolds.
result Convex sets in generic Riemannian manifolds are strictly convex if bounded by smooth hypersurfaces.
Recently, based on the idea of randomizing space theory, random convex analysis has been being developed in order to deal with the corresponding problems in random environments such as analysis of conditional convex risk measures and the related variational problems and optimization problems. Random convex analysis is …
As a generalization of geodesic function, in the present paper, we introduce the notion of geodesic φ-convex function and deduce some basic properties of φ-convex function and geodesic φ-convex function. We also introduce the concept of geodesic φ-convex set and φ-epigraph and in…
The paper shows that g-convex functions on manifolds are sparse.
problem Characterizing and understanding the sparseness of g-convex functions.
method Established criteria for g-convexity and used them to prove sparseness results.
result Most g-convex functions on compact manifolds have few critical points.
Finding efficient and provable methods to solve non-convex optimization problems is an outstanding challenge in machine learning and optimization theory. A popular approach used to tackle non-convex problems is to use convex relaxation techniques to find a convex surrogate for the problem. Unfortunately, convex relaxat…
Geodesic convexity generalizes the notion of (vector space) convexity to nonlinear metric spaces. But unlike convex optimization, geodesically convex (g-convex) optimization is much less developed. In this paper we contribute to the understanding of g-convex optimization by developing iteration complexity analysis for …
New method for optimization on Hadamard manifolds with curvature-independent guarantees.
problem Curvature-dependent complexity in geodesic convex optimization.
method Introducing horospherical convexity and developing algorithms for optimization.
result Curvature-independent convergence of subgradient descent and Nesterov's method.
Asymmetric expansion preserves convexity in hyperbolic geometry.
problem Maintaining convexity in hyperbolic geometry under asymmetric expansions.
method Generalizing earlier results on radial expansion to asymmetric expansion.
result Asymmetric expansion of hyperbolic convex sets remains convex.
Proof shows local convexity implies global convexity in special geometric spaces.
problem Proving convexity in CAT(0) cubed complexes from local convexity.
method Analyzes vertex link structures to determine convexity.
result Local combinatorial properties determine global convexity.
Flow deforms locally convex curves to curves of constant k-order width.
problem Evolve locally convex curves to curves of constant k-order width.
method Introduced a nonlocal curvature flow to evolve locally convex curves in the plane.
result The flow converges to a smooth, locally convex curve of constant k-order width as time goes to infinity.
Let U⊆Rd be open and convex. We prove that every (not necessarily Lipschitz or strongly) convex function f:U→R can be approximated by real analytic convex functions, uniformly on all of U. We also show that C0-fine approximation of convex functions by smooth (or real analytic) conv…
Convexity properties are preserved under radial transformations in hyperbolic and spherical geometries.
problem Preserving convexity in hyperbolic and spherical geometries under radial transformations.
method Used Poincaré disk model for hyperbolic geometry and stereographic projection for spherical geometry to prove preservation of convexity under radial expansion and contraction.
result Radial expansion and contraction preserve hyperbolic and spherical convexity, respectively.
Classifies geodetically convex sets and functions on Heisenberg group.
problem Characterizing geodetically convex sets and functions in the Heisenberg group.
method Classification through mathematical analysis.
result Geodetically convex sets and functions defined on Heisenberg group Hn classified. Convex optimization with sparsity-promoting convex regularization is a standard approach for estimating sparse signals in noise. In order to promote sparsity more strongly than convex regularization, it is also standard practice to employ non-convex optimization. In this paper, we take a third approach. We utilize a no…
Convex optimization models predict outputs from inputs via optimization problems.
problem Predicting outputs from inputs using convex optimization models.
method Proposed a heuristic for learning parameters of convex optimization models from datasets.
result Demonstrated the effectiveness of the proposed method on three model classes.
New algorithm improves convergence for non-convex problems with boundaries.
problem Optimizing non-convex problems with constraints.
method Reflected Gradient Langevin Dynamics with probabilistic representation.
result Promising convergence rates, faster than existing methods.
Unified framework for robust risk measures beyond convexity.
problem Developing risk measures for uncertainty beyond classical convexity.
method Constructing robust quasi-convex measures through uncertainty sets.
result Unified framework for robust quasi-convex risk measures.
AGGLIO optimizes non-convex functions with local convexity guarantees.
problem Optimizing non-convex functions with local convexity.
method Stage-wise, graduated optimization technique for locally convex functions.
result Global convergence to the global optimum for non-convex and locally convex objectives.
New illumination bodies defined for ball-convex shapes, proving convexity and establishing surface area measures.
problem Characterizing properties of ball-convex shapes.
method Introducing illumination bodies and weighted illumination bodies, proving convexity, and establishing surface area measures.
result Illumination bodies are convex and provide surface area measures for ball-convex shapes.
Sharp estimates for Finsler metrics in convex domains.
problem Estimating distances in Finsler metrics near convex points.
method Sharp estimates for intrinsic distances of Finsler metrics.
result Characterization of k-quasi hyperbolic metric in convex geometry. Strongly convex bodies can be approximated by smooth ones.
problem Approximating strongly convex bodies with smooth ones.
method Using C2 locally strongly convex bodies. result Smooth approximations of strongly convex bodies exist and can be controlled in terms of Hausdorff distance.
New index characterizes non-smooth Zoll convex bodies.
problem Characterizing non-smooth Zoll convex bodies.
method Defining systolic S1-index and using it to introduce generalized Zoll convex bodies. result Generalized Zoll convex bodies coincide with classical ones under certain conditions.
Study on convex ordering in stochastic control for swing contracts, proving value function convexity.
problem Pricing of swing contracts under stochastic dynamics.
method Discrete-time stochastic optimal control problem, convexity propagation, Brownian diffusion model, Stein's formula.
result Value function is convex in underlying asset price, relaxation of convexity assumption for semi-convexity.
Let U⊆Rn be open and convex. We show that every (not necessarily Lipschitz or strongly) convex function f:U→R can be approximated by real analytic convex functions, uniformly on all of U. In doing so we provide a technique which transfers results on uniform approximation on bounded …
The paper extends Stone duality to topological convexity spaces.
problem Understanding the relationship between topological convexity spaces and sup-lattices.
method Extending Stone duality to topological convexity spaces using preconvexity spaces.
result An adjunction between topological convexity spaces and sup-lattices.
Uniform convexity in divisible domains leads to hyperbolic geometry.
problem Understanding the geometry of divisible convex sets in Finsler manifolds.
method Proving β-uniform convexity of a specific Finsler metric. result A strictly convex divisible domain induces a β-uniformly convex Finsler metric. We show that C0-fine approximation of convex functions by smooth (or real analytic) convex functions on Rd is possible in general if and only if d=1. Nevertheless, for d≥2 we give a characterization of the class of convex functions on Rd which can be approximated by real analytic (or just smoother) c…
Generalizes smoothness conditions for optimization methods.
problem Optimization under non-uniform smoothness conditions.
method Develops a new analysis technique for bounding gradients.
result Obtains convergence rates for gradient descent and Nesterov's method.
We introduce and study a new class of $\eps$-convex bodies (extending the class of convex bodies) in metric and normed linear spaces. We analyze relations between characteristic properties of convex bodies, demonstrate how $\eps$-convex bodies connect with some classical results of Convex Geometry, as Helly theorem, an…
A theorem of Tits - Vinberg allows to build an action of a Coxeter group Γ on a properly convex open set Ω of the real projective space, thanks to the data P of a polytope and reflection across its facets. We give sufficient conditions for such action to be of finite covolume, convex-cocompact or geometrically fi…
Solves equality case in isoperimetric inequality for non-convex domains.
problem Equality case in relative isoperimetric inequality outside convex sets.
method Analyzes non-convex domains to settle the equality case.
result Solves the equality case for relative isoperimetric inequality outside arbitrary convex sets.
Convexity and convex functions play an important role in theoretical physics. To initiate a study of the possible uses of convex functions in General Relativity, we discuss the consequences of a spacetime (M,gμν) or an initial data set (Σ,hij,Kij) admitting a suitably defined convex function. We show how…
Convexity and convex functions play an important role in theoretical physics. To initiate a study of the possible uses of convex functions in General Relativity, we discuss the consequences of a spacetime (M,gμν) or an initial data set (Σ,hij,Kij) admitting a suitably defined convex function. We show how…
Boundary of fiber convex domains is a cohomological sphere.
problem Understanding the boundary properties of fiber convex domains.
method Analyzing smooth fiber convex domains with smooth boundaries.
result The boundary is a cohomological sphere.
This paper studies quasar-convex functions to improve optimization methods.
problem Improving optimization methods for non-convex functions.
method Study of first order methods for quasar-convex functions.
result Proves complexity upper bounds similar to convex functions.