Nonconvex minimization problems

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DOI10.1090/S0273-0979-1979-14595-6zbMath0441.49011WikidataQ55879775 ScholiaQ55879775MaRDI QIDQ3883068

Ivar Ekeland

Publication date: 1979

Published in: Bulletin of the American Mathematical Society (Search for Journal in Brave)

zbMATH Keywords

variational problem Hopf-Rinow theorem fixed point theorem nonlinear semigroup Pontryagin maximum principle mean-value theorem Asplund space Frechet derivative

Mathematics Subject Classification ID

Nonlinear programming (90C30) Normed linear spaces and Banach spaces; Banach lattices (46B99) Nonlinear parabolic equations (35K55) Fréchet and Gateaux differentiability in optimization (49J50) Equations involving nonlinear operators (general) (47J05) Numerical methods based on nonlinear programming (49M37) Fixed-point theorems (47H10) Control problems involving ordinary differential equations (34H05) Existence theories for optimal control problems involving ordinary differential equations (49J15) Riemannian, Finsler and other geometric structures on infinite-dimensional manifolds (58B20) Existence theories for problems in abstract spaces (49J27) Differentiation theory (Gateaux, Fréchet, etc.) on manifolds (58C20) Control/observation systems governed by ordinary differential equations (93C15) Optimality conditions for problems involving ordinary differential equations (49K15) Research exposition (monographs, survey articles) pertaining to calculus of variations and optimal control (49-02)

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elliptic problems in unbounded domains ⋮ The Pontryagin maximum principle for minimax problems of optimal control ⋮ Feedback minimum principle for quasi-optimal processes of terminally-constrained control problems ⋮ Ekeland's principle for cyclically antimonotone equilibrium problems ⋮ Takahashi's minimization theorem and some related results in quasi-metric spaces ⋮ Nondifferentiable optimization by smooth approximations ⋮ Multiplicity and concentration results for fractional Schrödinger system with steep potential wells ⋮ Necessary conditions of optimal impulse controls for distributed parameter systems ⋮ On the continuity of the polyconvex, quasiconvex and rank-oneconvex envelopes with respect to growth condition ⋮ Critical point theory for nondifferentiable functionals and applications ⋮ Ekeland-Type Variational Principles and Equilibrium Problems ⋮ Existence of solutions and optimal control for reflecting stochastic differential equations with applications to population control theory^* ⋮ Brezis-Browder Principle and Dependent Choice ⋮ Smooth bump functions and geomentry of Banach spaces ⋮ On generalized ordering principles in nonlinear analysis ⋮ Approximate subdifferentials and applications 3: the metric theory ⋮ Ekeland's variational principle and the mountain pass lemma ⋮ Detecting the completeness of a Finsler manifold via potential theory for its infinity Laplacian ⋮ _∊-gradients pour les ponctions localements lipschitziennes et applications ⋮ Tangent vectors to sets in the theory of geodesics ⋮ Necessary and sufficient conditions for optimal controls in viscous flow problems ⋮ The drop theorem, the petal theorem and Ekeland's variational principle ⋮ Critical points and nonlinear variational problems ⋮ On metric regularity of multifunctions ⋮ Subtangents with controlled slope ⋮ Pseudometric Extensions of the Brezis-Browder Ordering Principle ⋮ Minimal elements for product orders ⋮ A Characterization of Complete Metric Spaces ⋮ Existence of a positive solution for some quasilinear elliptic equations in \(\mathbb{R}^N\) ⋮ Existence and boundedness of a minimiser for a constrained minimisation problem on Rⁿ with limiting exponent ⋮ Density of the Range of Potential Operators ⋮ Regularisations of convex functions and slicewise suprema ⋮ Ekeland's variational principle for interval-valued functions ⋮ Ćirić's and Fisher's quasi-contractions in the framework of \(wt\)-distance ⋮ A generalization of ekellandz’s variational principle ⋮ On classes of subdifferentiability spaces of Ioffe ⋮ Unnamed Item ⋮ On the local surjection property ⋮ Kirchhoff-type fractional Laplacian problems with critical and singular nonlinearities ⋮ ₂-SYMMETRIC CRITICAL POINT THEOREMS FOR NON-DIFFERENTIABLE FUNCTIONS ⋮ Some existence and uniqueness theorems on ordered metric spaces via generalized distances ⋮ An ε-lagrange multiplier rule for a mathematical programming problem on banacch spaces^∗ ⋮ Linear and nonlinear error bounds for lower semicontinuous functions ⋮ On a system of nonlinear Schrödinger equations with quadratic interaction ⋮ A random variational principle with application to weak Hadamard differentiability of convex integral functionals ⋮ Solvability of Quasilinear Elliptic Equations with Nonlinear Boundary Conditions ⋮ On the mean value theorem ⋮ Function Variational Principles and Normed Minimizers ⋮ Functional variational principles and coercivity over normed spaces ⋮ Morse theory and existence of periodic solutions of convex hamiltonian systems ⋮ Existence and bifurcation for some elliptic problems on exterior strip domains ⋮ Multiple positive solutions of nonhomogeneous elliptic equations in unbounded domains ⋮ Eigenvalue problems and bifurcation of nonhomogeneous semilinear elliptic equations in exterior strip domains ⋮ A multiplier rule on a metric space ⋮ On maximal element theorems, variants of Ekeland's variational principle and their applications ⋮ On the relation between the weak Palais-Smale condition and coercivity given by Zhong ⋮ On the stochastic maximum principle in optimal control of degenerate diffusions with Lipschitz coefficients ⋮ Systems of equilibrium problems with applications to new variants of Ekeland's variational principle, fixed point theorems and parametric optimization problems ⋮ Integration of primal lower nice functions in Hilbert spaces ⋮ Evolution equations with lack of convexity ⋮ Weak contractor directions and weak directional contractions ⋮ Open mappings theorems and linearization stability ⋮ Existence of multiple positive solutions of subcritical semilinear elliptic problems in exterior strip domains ⋮ The strong Ekeland variational principle ⋮ Large deviation for diffusions and Hamilton-Jacobi equation in Hilbert spaces ⋮ A Variational approach to superlinear elliptic problems ⋮ On Supportless Convex Sets ⋮ Critical points for nondifferentiable functions in presence of splitting ⋮ Local wellposedness of constrained problems ⋮ ŁOJASIEWICZ-TYPE INEQUALITIES AND GLOBAL ERROR BOUNDS FOR NONSMOOTH DEFINABLE FUNCTIONS IN O-MINIMAL STRUCTURES ⋮ Singular elliptic problems with lack of compactness ⋮ The palais-smale condition versus coercivity ⋮ The variational principle and stochastic optimal control ⋮ Optimal control of variational inequalities with delays in the highest order spatial derivatives ⋮ The existence and regularity of multiple solutions for a class of infinitely degenerate elliptic equations ⋮ A variational principle and best proximity points ⋮ Sequentially lower complete spaces and Ekeland's variational principle ⋮ A Smooth Variational Principle With Applications to Subdifferentiability and to Differentiability of Convex Functions ⋮ Existence theorems for nonlinear operator equations without continuity condition ⋮ Binding of atoms and stability of molecules in hartree and thomas-fermi type theories. ⋮ Drop theorems and lipschitzianness tests via maximality procedures ⋮ Bounds for some minimizing sequences of functionals ⋮ Existence of multiple positive solutions of semilinear elliptic boundary value problems in the half space ⋮ A Hölder continuity result for a class of obstacle problems under non standard growth conditions ⋮ Differential inequalities via maximal element techniques ⋮ Nonsmooth Analysis: Differential Calculus of Nondifferentiable Mappings ⋮ The subgradient in n ⋮ The subdifferential in Banach spaces ⋮ Binding of atoms and stability of molecules in hartree and thomas—fermi type theories ⋮ Variational principles with generalized distances and the modelization of organizational change ⋮ Pontryagin's Principle of Mixed Control-State Constrained Optimal Control Governed by Fluid Dynamic Systems ⋮ Stability criteria for contractive semigroups via maximality procedures ⋮ ŁOJASIEWICZ INEQUALITY ON NON-COMPACT DOMAINS AND SINGULARITIES AT INFINITY ⋮ Subdifferentials are locally maximal monotone ⋮ Binding of atoms and stability of molecules in hartree and thomas-fermi type theories. ⋮ A generalization of the Brézis-Browder principle on ordered sets ⋮ An application of Ekeland's theorem to minimax problems

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