IEEE Transactions on Neural Networks and Learning Systems, ISSN 2162-237X, 10/2018, Volume 29, Issue 10, pp. 5008 - 5019

Online learning has been successfully applied in various machine learning problems. Conventional analysis of online learning achieves a sharp generalization...

Learning systems | Algorithm design and analysis | Machine learning algorithms | quadratic growth condition (QGC) | strong convexity | Convex functions | Acceleration | Convergence | Logistics | Generalization ability | online learning | REGRESSION | COMPUTER SCIENCE, HARDWARE & ARCHITECTURE | NEURAL-NETWORKS | COMPUTER SCIENCE, THEORY & METHODS | SELECTION | COMPUTER SCIENCE, ARTIFICIAL INTELLIGENCE | ENGINEERING, ELECTRICAL & ELECTRONIC | Generalization (Psychology) - physiology | Algorithms | Education, Distance - methods | Machine Learning | Humans | Neural Networks (Computer) | Learning | Learning algorithms | Distance learning | Machine learning | Convexity | Internet | Martingales | Descent | quadratic growth condition | generalization ability

Learning systems | Algorithm design and analysis | Machine learning algorithms | quadratic growth condition (QGC) | strong convexity | Convex functions | Acceleration | Convergence | Logistics | Generalization ability | online learning | REGRESSION | COMPUTER SCIENCE, HARDWARE & ARCHITECTURE | NEURAL-NETWORKS | COMPUTER SCIENCE, THEORY & METHODS | SELECTION | COMPUTER SCIENCE, ARTIFICIAL INTELLIGENCE | ENGINEERING, ELECTRICAL & ELECTRONIC | Generalization (Psychology) - physiology | Algorithms | Education, Distance - methods | Machine Learning | Humans | Neural Networks (Computer) | Learning | Learning algorithms | Distance learning | Machine learning | Convexity | Internet | Martingales | Descent | quadratic growth condition | generalization ability

Journal Article

Journal of Functional Analysis, ISSN 0022-1236, 03/2019, Volume 276, Issue 6, pp. 1806 - 1852

We consider fully nonlinear uniformly elliptic equations with quadratic growth in the gradient, such as−F(x,u,Du,D2u)=λc(x)u+〈M(x)Du,Du〉+h(x) in a bounded...

A priori bounds | Quadratic growth | Fully nonlinear | Multiplicity | EXISTENCE | MATHEMATICS | VISCOSITY SOLUTIONS | MAXIMUM PRINCIPLE | REGULARITY | ELLIPTIC-EQUATIONS

A priori bounds | Quadratic growth | Fully nonlinear | Multiplicity | EXISTENCE | MATHEMATICS | VISCOSITY SOLUTIONS | MAXIMUM PRINCIPLE | REGULARITY | ELLIPTIC-EQUATIONS

Journal Article

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Adaptive restart of accelerated gradient methods under local quadratic growth condition

IMA Journal of Numerical Analysis, ISSN 0272-4979, 10/2019, Volume 39, Issue 4, pp. 2069 - 2095

Abstract By analyzing accelerated proximal gradient methods under a local quadratic growth condition, we show that restarting these algorithms at any frequency...

MATHEMATICS, APPLIED | restarting | COORDINATE DESCENT METHODS | quadratic growth condition | MINIMIZATION | ALGORITHM | CONVERGENCE | accelerated gradient descent | unknown error bound | PARALLEL | Mathematics - Optimization and Control | Mathematics | Optimization and Control

MATHEMATICS, APPLIED | restarting | COORDINATE DESCENT METHODS | quadratic growth condition | MINIMIZATION | ALGORITHM | CONVERGENCE | accelerated gradient descent | unknown error bound | PARALLEL | Mathematics - Optimization and Control | Mathematics | Optimization and Control

Journal Article

Journal of Functional Analysis, ISSN 0022-1236, 02/2019, Volume 276, Issue 4, pp. 1294 - 1312

In this paper, we study semilinear elliptic systems with critical nonlinearity of the form(0.1)Δu=Q(x,u,∇u), for u:Rn→RK, Q has quadratic growth in ∇u. Our...

Quadratic growth of gradient | Elliptic systems with critical nonlinearity | Harmonic and biharmonic maps | Lorentz space | MATHEMATICS | HARMONIC MAPS | nonlinearity | Elliptic systems with critical

Quadratic growth of gradient | Elliptic systems with critical nonlinearity | Harmonic and biharmonic maps | Lorentz space | MATHEMATICS | HARMONIC MAPS | nonlinearity | Elliptic systems with critical

Journal Article

Journal of Mathematical Analysis and Applications, ISSN 0022-247X, 04/2017, Volume 448, Issue 2, pp. 1120 - 1146

In this paper, we deal with the following singular elliptic system:{−Δu+α|∇u|2u=pp+qa(x)|v|q|u|p−2u+f,x∈RN,−Δv+β|∇v|2v=qp+qa(x)|u|p|v|q−2v+g,x∈RN, where N≥3,...

Singular elliptic systems | Sub-supersolutions | Concentration-compactness principle | Quadratic growth in the gradient | Variational method | Nehari manifold | EXISTENCE | MATHEMATICS, APPLIED | NONEXISTENCE | QUASI-LINEAR EQUATIONS | CONVECTION TERM | MATHEMATICS | SYMMETRY | POSITIVE ENTIRE SOLUTIONS | DOMAINS

Singular elliptic systems | Sub-supersolutions | Concentration-compactness principle | Quadratic growth in the gradient | Variational method | Nehari manifold | EXISTENCE | MATHEMATICS, APPLIED | NONEXISTENCE | QUASI-LINEAR EQUATIONS | CONVECTION TERM | MATHEMATICS | SYMMETRY | POSITIVE ENTIRE SOLUTIONS | DOMAINS

Journal Article

Journal of Functional Analysis, ISSN 0022-1236, 04/2015, Volume 268, Issue 8, pp. 2298 - 2335

We consider the boundary value problem(Pλ)u∈H01(Ω)∩L∞(Ω):−Δu=λc(x)u+μ(x)|∇u|2+h(x), where Ω⊂RN, N≥3 is a bounded domain with smooth boundary. It is assumed...

Quadratic growth in the gradient | Topological degree | Continuum of solutions | Elliptic equations | EXISTENCE | MATHEMATICS | Topological | QUADRATIC GROWTH | EQUATIONS | UNIQUENESS | Analysis of PDEs | Mathematics

Quadratic growth in the gradient | Topological degree | Continuum of solutions | Elliptic equations | EXISTENCE | MATHEMATICS | Topological | QUADRATIC GROWTH | EQUATIONS | UNIQUENESS | Analysis of PDEs | Mathematics

Journal Article

Journal de mathématiques pures et appliquées, ISSN 0021-7824, 12/2019, Volume 132, pp. 308 - 333

Let Ω⊂RN, N≥2, be a smooth bounded domain. We consider a boundary value problem of the form−Δu=cλ(x)u+μ(x)|∇u|2+h(x),u∈H01(Ω)∩L∞(Ω) where cλ depends on a...

A priori bound | p-Laplacian | Critical growth in the gradient | Boundary weak Harnack inequality | Continuum of solutions | EXISTENCE | MATHEMATICS | MATHEMATICS, APPLIED | QUADRATIC GROWTH | EQUATIONS | UNIQUENESS | Analysis of PDEs | Mathematics

A priori bound | p-Laplacian | Critical growth in the gradient | Boundary weak Harnack inequality | Continuum of solutions | EXISTENCE | MATHEMATICS | MATHEMATICS, APPLIED | QUADRATIC GROWTH | EQUATIONS | UNIQUENESS | Analysis of PDEs | Mathematics

Journal Article

Journal of Differential Equations, ISSN 0022-0396, 05/2017, Volume 262, Issue 10, pp. 5231 - 5270

We consider the boundary value problem(Pλ)−Δu=λc(x)u+μ(x)|∇u|2+h(x),u∈H01(Ω)∩L∞(Ω), where Ω⊂RN,N≥3 is a bounded domain with smooth boundary. It is assumed that...

Lower and upper solutions | Quasilinear elliptic equations | Quadratic growth in the gradient | MATHEMATICS | MAXIMUM PRINCIPLE | QUADRATIC GROWTH | EQUATIONS | UNIQUENESS | Analysis of PDEs | Mathematics

Lower and upper solutions | Quasilinear elliptic equations | Quadratic growth in the gradient | MATHEMATICS | MAXIMUM PRINCIPLE | QUADRATIC GROWTH | EQUATIONS | UNIQUENESS | Analysis of PDEs | Mathematics

Journal Article

SIAM Journal on Control and Optimization, ISSN 0363-0129, 2015, Volume 53, Issue 1, pp. 185 - 212

This paper studies the large time behavior of solutions to semilinear Cauchy problems with quadratic nonlinearity in gradients. The Cauchy problem considered...

Ergodic equation | Quadratic growth gradient | Large time behavior | Semilinear equation | MATHEMATICS, APPLIED | quadratic growth gradient | RISK-SENSITIVE CONTROL | PDES | semilinear equation | large time behavior | BELLMAN EQUATIONS | ergodic equation | ERGODIC BSDES | AUTOMATION & CONTROL SYSTEMS | Mathematical analysis | Differential equations | Nonlinearity | Boundaries | Stochasticity | Optimization | Convergence | Cauchy problem

Ergodic equation | Quadratic growth gradient | Large time behavior | Semilinear equation | MATHEMATICS, APPLIED | quadratic growth gradient | RISK-SENSITIVE CONTROL | PDES | semilinear equation | large time behavior | BELLMAN EQUATIONS | ergodic equation | ERGODIC BSDES | AUTOMATION & CONTROL SYSTEMS | Mathematical analysis | Differential equations | Nonlinearity | Boundaries | Stochasticity | Optimization | Convergence | Cauchy problem

Journal Article

Journal of Differential Equations, ISSN 0022-0396, 01/2014, Volume 256, Issue 2, pp. 577 - 608

In this paper we consider the problem{u∈W01,p(Ω),−diva(x,u,Du)+b(x,u,Du)=h(x,u,Du)in D′(Ω), where −diva(x,u,Du) is a Leray–Lions operator which is defined on...

Existence | Nonlinear elliptic equations | MATHEMATICS | BOUNDED SOLUTIONS | EXISTENCE RESULT | QUADRATIC GROWTH | Analysis of PDEs | Mathematics

Existence | Nonlinear elliptic equations | MATHEMATICS | BOUNDED SOLUTIONS | EXISTENCE RESULT | QUADRATIC GROWTH | Analysis of PDEs | Mathematics

Journal Article

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Existence and Multiplicity for Elliptic Problems with Quadratic Growth in the Gradient

Communications in Partial Differential Equations, ISSN 0360-5302, 02/2013, Volume 38, Issue 2, pp. 244 - 264

We show that a class of divergence-form elliptic problems with quadratic growth in the gradient and non-coercive zero order terms are solvable, under...

Non-coercive | Sub- and super-solutions | Natural growth | Quadratic growth in the gradient | Elliptic equation | Variational methods | MATHEMATICS, APPLIED | 35J62 | EQUATIONS | 35J25 | UNIQUENESS | SOURCE TERMS | MATHEMATICS | ORDER | Studies | Partial differential equations | Algorithms | Hypotheses | Mathematical analysis | Optimization | Uniqueness | Mathematics - Analysis of PDEs | Analysis of PDEs | Mathematics

Non-coercive | Sub- and super-solutions | Natural growth | Quadratic growth in the gradient | Elliptic equation | Variational methods | MATHEMATICS, APPLIED | 35J62 | EQUATIONS | 35J25 | UNIQUENESS | SOURCE TERMS | MATHEMATICS | ORDER | Studies | Partial differential equations | Algorithms | Hypotheses | Mathematical analysis | Optimization | Uniqueness | Mathematics - Analysis of PDEs | Analysis of PDEs | Mathematics

Journal Article

NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS, ISSN 0362-546X, 07/2015, Volume 121, pp. 412 - 423

We study nonnegative solutions of the boundary value problem -Delta u=lambda c(x)u + mu(x)vertical bar del u vertical bar(2) + h(x), u is an element of H-0(1)...

MATHEMATICS, APPLIED | SPACES | Bifurcation | Weighted Sobolev inequalities | Hardy inequalities | MATHEMATICS | L-infinity(p) spaces | Elliptic equation | EXISTENCE RESULT | Critical growth in the gradient | QUADRATIC GROWTH | A priori estimates | Multiplicity | BOUNDED SOLUTIONS | Existence | Boundary value problems | Infinity | Mathematical analysis | Continuums | Texts | Projection | Bifurcations | Estimates

MATHEMATICS, APPLIED | SPACES | Bifurcation | Weighted Sobolev inequalities | Hardy inequalities | MATHEMATICS | L-infinity(p) spaces | Elliptic equation | EXISTENCE RESULT | Critical growth in the gradient | QUADRATIC GROWTH | A priori estimates | Multiplicity | BOUNDED SOLUTIONS | Existence | Boundary value problems | Infinity | Mathematical analysis | Continuums | Texts | Projection | Bifurcations | Estimates

Journal Article

DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS, ISSN 1078-0947, 05/2019, Volume 39, Issue 5, pp. 2637 - 2659

In this paper, we establish the existence of a positive solution to {-M lambda,Lambda(+)(D(2)u) + H (x, Du) = K(x)f(u)/u(alpha) in Omega, u > 0 in Omega, u = 0...

EXISTENCE | VISCOSITY SOLUTIONS | MATHEMATICS, APPLIED | MAXIMUM PRINCIPLE | NONEXISTENCE | CONVECTION | PRINCIPAL EIGENVALUES | MATHEMATICS | ELLIPTIC PROBLEMS | viscosity solution | Pucci's extremal operator | positive solution | singular and gradient nonlinearity | QUADRATIC GROWTH | DIRICHLET PROBLEM | BIFURCATION

EXISTENCE | VISCOSITY SOLUTIONS | MATHEMATICS, APPLIED | MAXIMUM PRINCIPLE | NONEXISTENCE | CONVECTION | PRINCIPAL EIGENVALUES | MATHEMATICS | ELLIPTIC PROBLEMS | viscosity solution | Pucci's extremal operator | positive solution | singular and gradient nonlinearity | QUADRATIC GROWTH | DIRICHLET PROBLEM | BIFURCATION

Journal Article

Journal of Mathematical Analysis and Applications, ISSN 0022-247X, 2009, Volume 350, Issue 1, pp. 401 - 408

Given a bounded domain Ω in R N , and a function a ∈ L q ( Ω ) with q > N / 2 , we study the existence of a positive solution for the quasilinear problem − Δ w...

Singular quasilinear elliptic equations | Positive solutions | Quadratic growth in the gradient | EXISTENCE | MATHEMATICS | MATHEMATICS, APPLIED | NONLINEAR ELLIPTIC-EQUATIONS | NATURAL GROWTH | TERMS

Singular quasilinear elliptic equations | Positive solutions | Quadratic growth in the gradient | EXISTENCE | MATHEMATICS | MATHEMATICS, APPLIED | NONLINEAR ELLIPTIC-EQUATIONS | NATURAL GROWTH | TERMS

Journal Article

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Existence of solutions of sublinear elliptic systems with quadratic growth in the gradient

Nonlinear Differential Equations and Applications NoDEA, ISSN 1021-9722, 10/2014, Volume 21, Issue 5, pp. 737 - 749

In this paper, we deal with the following sublinear elliptic system: $${{\left\{\begin{array}{ll}-\Delta u + u + |\nabla u|^2 = a(x)|v|^p+ f,\quad x \in...

Nonnegative solutions | 35J60 | Analysis | Mathematics | Quadratic growth in the gradient | Sublinear elliptic systems | 35J55 | MATHEMATICS, APPLIED | NONEXISTENCE | SYMMETRY | POSITIVE ENTIRE SOLUTIONS | EQUATIONS | CONVECTION TERM | UNIQUENESS

Nonnegative solutions | 35J60 | Analysis | Mathematics | Quadratic growth in the gradient | Sublinear elliptic systems | 35J55 | MATHEMATICS, APPLIED | NONEXISTENCE | SYMMETRY | POSITIVE ENTIRE SOLUTIONS | EQUATIONS | CONVECTION TERM | UNIQUENESS

Journal Article

数学物理学报：B辑英文版, ISSN 0252-9602, 2015, Volume 35, Issue 5, pp. 1023 - 1036

In this article, we consider existence and nonexistence of solutions to problem {-△pu＋g（x,u）｜↓△｜^p=f in -Ω,u=0 on Ω with 1〈p〈∞ where f is a positive measurable...

解的存在性 | 梯度 | 开集 | 非线性椭圆型方程 | 有界 | 35D05 | quasilinear elliptic equations | existence and nonexistence | 35D10 | 35J92 | 46E30 | gradient terms | singular weights | Existence and nonexistence | Singular weights | Quasilinear elliptic equations | Gradient terms | EXISTENCE | MATHEMATICS | P-GROWTH | QUADRATIC GROWTH | QUASI-LINEAR EQUATIONS | NATURAL GROWTH TERMS

解的存在性 | 梯度 | 开集 | 非线性椭圆型方程 | 有界 | 35D05 | quasilinear elliptic equations | existence and nonexistence | 35D10 | 35J92 | 46E30 | gradient terms | singular weights | Existence and nonexistence | Singular weights | Quasilinear elliptic equations | Gradient terms | EXISTENCE | MATHEMATICS | P-GROWTH | QUADRATIC GROWTH | QUASI-LINEAR EQUATIONS | NATURAL GROWTH TERMS

Journal Article

ESAIM: Control, Optimisation and Calculus of Variations, ISSN 1292-8119, 7/2008, Volume 14, Issue 3, pp. 411 - 426

We present a revisited form of a result proved in [Boccardo, Murat and Puel, Portugaliae Math. 41 ( 1982) 507-534] and then we adapt the new proof in order to...

Singular lower order term | Quadratic gradient | EXISTENCE | MATHEMATICS, APPLIED | PARABOLIC EQUATIONS | NONLINEAR ELLIPTIC-EQUATIONS | NATURAL GROWTH TERMS | BOUNDED SOLUTIONS | quadratic gradient | singular lower order term | AUTOMATION & CONTROL SYSTEMS

Singular lower order term | Quadratic gradient | EXISTENCE | MATHEMATICS, APPLIED | PARABOLIC EQUATIONS | NONLINEAR ELLIPTIC-EQUATIONS | NATURAL GROWTH TERMS | BOUNDED SOLUTIONS | quadratic gradient | singular lower order term | AUTOMATION & CONTROL SYSTEMS

Journal Article

Atti della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali, Rendiconti Lincei Matematica e Applicazioni, ISSN 1120-6330, 2016, Volume 27, Issue 2, pp. 195 - 233

In this paper we consider the problem [GRAPHICS] where Omega is an open bounded set of R-N, N >= 3, Lambda(x) is a coercive matrix with coefficients in...

Perturbation with quadratic growth in the gradient | Noncoercive zeroth order term | Quasilinear problems | Existence | MATHEMATICS | MATHEMATICS, APPLIED | EXISTENCE RESULT | noncoercive zeroth order term | NONLINEAR ELLIPTIC-EQUATIONS | NATURAL GROWTH | existence | BOUNDED SOLUTIONS | perturbation with quadratic growth in the gradient | UNIQUENESS | Mathematics - Analysis of PDEs

Perturbation with quadratic growth in the gradient | Noncoercive zeroth order term | Quasilinear problems | Existence | MATHEMATICS | MATHEMATICS, APPLIED | EXISTENCE RESULT | noncoercive zeroth order term | NONLINEAR ELLIPTIC-EQUATIONS | NATURAL GROWTH | existence | BOUNDED SOLUTIONS | perturbation with quadratic growth in the gradient | UNIQUENESS | Mathematics - Analysis of PDEs

Journal Article

Asian Journal of Mathematics, ISSN 1093-6106, 2015, Volume 19, Issue 5, pp. 933 - 950

We study the geometry at infinity of expanding gradient Ricci solitons (M-n, g, del f), n >= 3, with finite asymptotic curvature ratio without curvature sign...

Asymptotic cone | Ricci flow | Expanding gradient ricci soliton | MATHEMATICS | VOLUME GROWTH | MATHEMATICS, APPLIED | expanding gradient Ricci soliton | asymptotic cone | QUADRATIC CURVATURE DECAY | CONVERGENCE | NONNEGATIVELY CURVED MANIFOLDS | FLOW | RIEMANNIAN-MANIFOLDS

Asymptotic cone | Ricci flow | Expanding gradient ricci soliton | MATHEMATICS | VOLUME GROWTH | MATHEMATICS, APPLIED | expanding gradient Ricci soliton | asymptotic cone | QUADRATIC CURVATURE DECAY | CONVERGENCE | NONNEGATIVELY CURVED MANIFOLDS | FLOW | RIEMANNIAN-MANIFOLDS

Journal Article

Journal of Mathematical Analysis and Applications, ISSN 0022-247X, 12/2014, Volume 420, Issue 1, pp. 772 - 780

In this note we present some uniqueness and comparison results for a class of problem of the form(0.1)−Lu=H(x,u,∇u)+h(x),u∈H01(Ω)∩L∞(Ω), where Ω⊂RN, N≥2 is a...

Uniqueness of solution | Quasilinear elliptic equations | Quadratic growth in the gradient | MATHEMATICS | MATHEMATICS, APPLIED

Uniqueness of solution | Quasilinear elliptic equations | Quadratic growth in the gradient | MATHEMATICS | MATHEMATICS, APPLIED

Journal Article

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