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. The paper introduces geodesic φ-convex functions and their properties.
problem Generalizing geodesic functions to φ-convex functions.
method Introducing geodesic φ-convex functions and investigating their properties.
result Characterization of geodesic φ-convex functions via their φ-epigraphs.
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. 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.
Convex functions restrict spacetime geometry in GR.
problem Understanding constraints on spacetime geometry.
method Analyzing spacetimes and initial data sets with convex functions.
result Existence of convex functions imposes geometric restrictions.
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…
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…
No non-trivial convex functions on certain Finsler manifolds.
problem Existence of convex functions on Finsler manifolds.
method Analyzing Holmes- Thompson volume and Busemann function properties.
result Non-compact Finsler manifolds with infinite Holmes- Thompson volume have no non-trivial convex functions.
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…
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 connects convex functions to p-subharmonic functions and proves their equivalence.
problem Understanding the relationship between convex functions and p-subharmonic functions.
method Average principle, variational methods, and PDE techniques.
result Convex functions on R^n are p-subharmonic for every p > 1.
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.
Characterizes convexity of distance functions on Riemannian manifolds.
problem Understanding convexity of distance functions on Riemannian manifolds.
method Characterization of proximal normal cones, separation theorems, and analysis of convex subsets' boundaries.
result Convexity of distance functions for various boundary conditions on Riemannian manifolds.
Least Squares Estimators are suboptimal for 5D convex functions.
problem Suboptimality of Least Squares Estimators in estimating multidimensional convex functions.
method Analysis of natural subclasses of convex functions in random and fixed design settings.
result Risk of LSE is n−2/d while minimax risk is n−4/(d+4) for d≥5. Paper proves non-existence of certain convex functions on a Riemannian manifold with a pole.
problem Proving non-existence of specific convex functions on a Riemannian manifold with a pole.
method Developed notions of odd and even functions on a Riemannian manifold with a pole, proved non-existence of non-trivial and non-negative convex functions.
result Deduced non-existence of non-trivial and non-negative differentiable odd convex functions whose gradient is complete.
Proves convexity of minimizers in energy functions with convex potentials.
problem Connectedness and convexity of minimizers in energy functions involving surface tensions and convex potentials.
method Introduces a 'two-point function' to measure lack of convexity and prove negative second variation of the energy.
result Positively answers an old question of Almgren about connectedness and convexity of minimizers.
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.
The study links Ricci curvature and convexity in complex tori.
problem Characterizing Ricci curvature signs in toric manifolds.
method Characterization through convexity of volume functional.
result Relationships between Ricci curvature, volume, submanifolds, and pluri-subharmonic functions.
Paper tackles non-convex inf-projection problems with stochastic optimization.
problem Non-convex and possibly non-smooth inf-projection minimization problems.
method Developed stochastic algorithms for finding (nearly) stationary solutions.
result Established first-order convergence for non-convex inf-projection problems.
Established strong geodesic convex functions and their properties.
problem Geodesic convex functions and monotone vector fields on Riemannian manifolds.
method Characterization and relation establishment for strong geodesic convex functions.
result Relation between variational inequality solutions and strict minimizers for multiobjective programming.
New neural network approximates convex option prices.
problem Approximating prices of options with convex payoffs.
method Input Convex Neural Network (ICNN) architecture, with a scrambling phase.
result Validated convergence and effectiveness in estimating option prices.
First order methods can take extremely long to find global minima of non-convex functions.
problem Finding global minimizers of non-convex functions.
method Designing a family of non-convex functions and using statistical lower bounds for parameter estimation.
result First order methods can take exponential time to converge to a global minimizer.
New methods achieve linear convergence on broader convex functions.
problem Optimization on broader convex functions with singular or unbounded second derivatives.
method Discretizations of conformal Hamiltonian dynamics.
result Linear convergence on convex functions with singular or unbounded second derivatives.
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…
New functions linked to curvature bounds in Lorentzian manifolds.
problem Curvature bounds in Lorentzian manifolds and their relation to convex functions.
method Established a connection between sectional curvature bounds and space-time convex and λ-convex functions. result Natural construction of space-time convex and λ-convex functions. 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.
The paper explores convex functions on Riemannian manifolds and their geometric properties.
problem Existence and non-existence of convex functions on Riemannian manifolds.
method Analyzes geometric properties and conditions for the existence of convex functions on Riemannian manifolds.
result Geometric conditions ensuring the existence of convex functions on certain manifolds.
Optimally shows the distance between perturbed convex functions and their Γ-regularizations.
problem Understanding the difference between perturbed convex functions and their Γ-regularizations.
method Analyzing the compactly supported perturbation and the Γ-regularization of a strictly convex function.
result The optimal estimate of the distance between perturbed convex functions and their Γ-regularizations is shown to be o(ε). 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. This paper addresses the problem of sparsity penalized least squares for applications in sparse signal processing, e.g. sparse deconvolution. This paper aims to induce sparsity more strongly than L1 norm regularization, while avoiding non-convex optimization. For this purpose, this paper describes the design and use of…
We show that domains, that allow for convex functions with unbounded gradient at their boundary, are convex.
Universal algorithm minimizes adaptive regret for various convex functions.
problem Minimizing adaptive regret in changing environments for multiple convex functions.
method Borrowing MetaGrad's idea of multiple learning rates and using sleeping experts.
result First universal algorithm for minimizing adaptive regret of convex functions.
Study beta function for convex billiard maps, linking spectral invariants.
problem Understanding spectral invariants of convex billiard maps.
method Birkhoff normal form via constructive generating functions, explicit beta function formula.
result Linked spectral invariants to beta function for convex billiard maps.
Convex functions and bodies can be approximated by smoother convex functions.
problem Approximating convex functions and bodies with smoother ones.
method Using properties of convex functions and bodies, constructing smoother approximations.
result Smooth approximations of convex functions and bodies exist for any given tolerance.
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.
Adaptive sampling improves convex function learning.
problem Learning convex functions in the L∞ norm. method Function-specific complexity measure for adaptive sampling.
result Adaptive sampling nearly achieves optimal error rate.
The paper proves abelian convexity theorems using Kempf-Ness functions.
problem Establishing convexity along orbits in general settings.
method Using Kempf-Ness functions to prove abelian convexity theorems.
result Short proofs of Atiyah-Guillemin-Sternberg theorem and abelian convexity for gradient maps.
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.
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 …
Constructs 2-convex functions approximating distances in Alexandrov spaces.
problem Distance approximation in finite-dimensional Alexandrov spaces.
method Constructs 2-convex functions in Alexandrov spaces.
result Functions can be lifted to close Alexandrov spaces.
This paper optimizes functions of probability measures using particle gradient descent for displacement convex functions.
problem Optimizing functions of probability measures with displacement convex properties.
method Particle gradient descent applied to displacement convex functions with theoretical guarantees.
result Finite number of particles and computations are sufficient to find optimal solutions for displacement convex functions.
Empirical risk minimization frequently employs convex surrogates to underlying discrete loss functions in order to achieve computational tractability during optimization. However, classical convex surrogates can only tightly bound modular loss functions, sub-modular functions or supermodular functions separately while …
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.
In the article the necessary and sufficient conditions for a representation of Lipschitz function of two variables as a difference of two convex functions are formulated. An algorithm of this representation is given. The outcome of this algorithm is a sequence of pairs of convex functions that converge uniformly to a p…
We study functions whose truncations are convex or quasiconvex.
problem Understanding functions with specific truncation properties.
method Analyzing C2-smooth functions with positive definite Hessians. result Injectivity of restricted gradient in positive definite region.
The paper explores different smooth map notions on convex sets and their relationships.
problem Exploring and comparing different smooth map notions on convex sets.
method Constructing a function that doesn't extend to a smooth function on any open neighborhood but does for Ck functions. result Diffeological and Sikorski smoothness notions do not coincide for all convex sets.
SAdam improves Adam's performance for strongly convex functions.
problem Improving Adam's performance for strongly convex functions.
method Developed SAdam, a variant of Adam, which exploits strong convexity.
result Achieves a data-dependent O(logT) regret bound for strongly convex functions. We find a different approach to define convex functions in the sub-Riemannian setting. A function on a sub-Riemannian manifold is nonholonomically geodesic convex if its restriction to any nonholonomic (straightest) geodesic is convex. In the case of Carnot groups, this definition coincides with that by Danniell-Garofa…