Computer methods in applied mechanics and engineering, ISSN 0045-7825, 2019, Volume 354, pp. 990 - 1026

.... In our study, the irreversibility is handled via penalization. Provided the penalty constant is well-tuned, the penalized formulation is a good approximation...

Phase-field formulation | Irreversibility condition | Explicit lower bound | Penalty constant | Brittle fracture | ENERGY | CRACK-PROPAGATION | FAILURE CRITERIA | FINITE-ELEMENT APPROXIMATION | NUMERICAL IMPLEMENTATION | FORMULATION | GRADIENT DAMAGE MODELS | BALANCE | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | ENGINEERING, MULTIDISCIPLINARY | DEGRADATION | Finite element method | Lower bounds | Exact solutions | Boundary conditions | Regularization | Optimization | Fracture toughness

Phase-field formulation | Irreversibility condition | Explicit lower bound | Penalty constant | Brittle fracture | ENERGY | CRACK-PROPAGATION | FAILURE CRITERIA | FINITE-ELEMENT APPROXIMATION | NUMERICAL IMPLEMENTATION | FORMULATION | GRADIENT DAMAGE MODELS | BALANCE | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | ENGINEERING, MULTIDISCIPLINARY | DEGRADATION | Finite element method | Lower bounds | Exact solutions | Boundary conditions | Regularization | Optimization | Fracture toughness

Journal Article

Optimization letters, ISSN 1862-4480, 2017, Volume 13, Issue 3, pp. 597 - 615

.... The original problem is reduced to a problem without inequality constraints by means of the exact penalization techniques...

Linearized problem | Mathematics | Inequality constraints | KKT point | Optimization | Computational Intelligence | Exact penalty | Global optimality conditions | D.c. functions | Operations Research/Decision Theory | Numerical and Computational Physics, Simulation | Constructive property | Lagrange function | MATHEMATICS, APPLIED | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | OPTIMIZATION | D.c. functions

Linearized problem | Mathematics | Inequality constraints | KKT point | Optimization | Computational Intelligence | Exact penalty | Global optimality conditions | D.c. functions | Operations Research/Decision Theory | Numerical and Computational Physics, Simulation | Constructive property | Lagrange function | MATHEMATICS, APPLIED | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | OPTIMIZATION | D.c. functions

Journal Article

Mathematical Programming, ISSN 0025-5610, 3/2018, Volume 168, Issue 1, pp. 571 - 598

...) conditions to hold at a local minimizer, let alone ensuring an exact penalization. In this paper, we extend quasi-normality and relaxed constant positive linear...

Theoretical, Mathematical and Computational Physics | Error bound | Mathematics | 90C26 | Exact penalization | Non-Lipschitz program | Mathematical Methods in Physics | 90C30 | 90C46 | Calculus of Variations and Optimal Control; Optimization | Mathematics of Computing | Numerical Analysis | Necessary optimality | Combinatorics | COMPUTER SCIENCE, SOFTWARE ENGINEERING | MATHEMATICS, APPLIED | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | CALMNESS | CONSTRAINT QUALIFICATIONS | SIGNALS | OPTIMIZATION | Composite functions | Continuity (mathematics) | Normality

Theoretical, Mathematical and Computational Physics | Error bound | Mathematics | 90C26 | Exact penalization | Non-Lipschitz program | Mathematical Methods in Physics | 90C30 | 90C46 | Calculus of Variations and Optimal Control; Optimization | Mathematics of Computing | Numerical Analysis | Necessary optimality | Combinatorics | COMPUTER SCIENCE, SOFTWARE ENGINEERING | MATHEMATICS, APPLIED | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | CALMNESS | CONSTRAINT QUALIFICATIONS | SIGNALS | OPTIMIZATION | Composite functions | Continuity (mathematics) | Normality

Journal Article

Optimal Control Applications and Methods, ISSN 0143-2087, 11/2016, Volume 37, Issue 6, pp. 1329 - 1354

.... We analyze the exact penalization of the terminal constraints. We show that for systems that are exactly controllable, the norm...

L1 optimal control | abstract Cauchy problems | terminal constraint | heat equation | exact penalization | wave equation | exact controllability | optimal control | method of moments | moment equations | nonsmooth optimization | L-1 optimal control | MATHEMATICS, APPLIED | WAVE | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | NORM-MINIMAL CONTROL | BOUNDARY CONTROL | CONTROLLABILITY | AUTOMATION & CONTROL SYSTEMS | Linear systems | Control methods | Mathematical analysis | Optimal control | Mathematical models | Terminals | Terminal constraints | Optimization

L1 optimal control | abstract Cauchy problems | terminal constraint | heat equation | exact penalization | wave equation | exact controllability | optimal control | method of moments | moment equations | nonsmooth optimization | L-1 optimal control | MATHEMATICS, APPLIED | WAVE | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | NORM-MINIMAL CONTROL | BOUNDARY CONTROL | CONTROLLABILITY | AUTOMATION & CONTROL SYSTEMS | Linear systems | Control methods | Mathematical analysis | Optimal control | Mathematical models | Terminals | Terminal constraints | Optimization

Journal Article

Journal of Optimization Theory and Applications, ISSN 0022-3239, 9/2014, Volume 162, Issue 3, pp. 856 - 872

In this paper, we study the relationship between calmness and exact penalization for vector optimization problems under nonlinear perturbations...

(Weakly) efficient solution | Value function | Calculus of Variations and Optimal Control; Optimization | Operations Research/Decision Theory | Mathematics | Theory of Computation | Applications of Mathematics | Engineering, general | Exact penalization | Vector optimization under nonlinear perturbations | Calmness | Optimization | MATHEMATICS, APPLIED | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | EXACT PENALTY-FUNCTIONS | Studies | Operations research | Mathematical models | Nonlinearity | Perturbation | Vectors (mathematics) | Mathematical analysis

(Weakly) efficient solution | Value function | Calculus of Variations and Optimal Control; Optimization | Operations Research/Decision Theory | Mathematics | Theory of Computation | Applications of Mathematics | Engineering, general | Exact penalization | Vector optimization under nonlinear perturbations | Calmness | Optimization | MATHEMATICS, APPLIED | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | EXACT PENALTY-FUNCTIONS | Studies | Operations research | Mathematical models | Nonlinearity | Perturbation | Vectors (mathematics) | Mathematical analysis

Journal Article

Journal of Optimization Theory and Applications, ISSN 0022-3239, 10/2016, Volume 171, Issue 1, pp. 228 - 250

In this paper, we deal with the error bounds for inequality systems and the exact penalization for constrained optimization problems...

Subdifferential | Concave | Error bound | Mathematics | Theory of Computation | 90C26 | Optimization | 90C31 | Exact penalty | 90C30 | Calculus of Variations and Optimal Control; Optimization | 49J52 | Applications of Mathematics | Engineering, general | Operation Research/Decision Theory | Quadratic | LOWER SEMICONTINUOUS FUNCTIONS | MATHEMATICS, APPLIED | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | Computer science | Studies | Quadratic programming | Quadratic equations | Errors | Constraints | Error analysis | Inequalities | Convexity | Computer Science

Subdifferential | Concave | Error bound | Mathematics | Theory of Computation | 90C26 | Optimization | 90C31 | Exact penalty | 90C30 | Calculus of Variations and Optimal Control; Optimization | 49J52 | Applications of Mathematics | Engineering, general | Operation Research/Decision Theory | Quadratic | LOWER SEMICONTINUOUS FUNCTIONS | MATHEMATICS, APPLIED | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | Computer science | Studies | Quadratic programming | Quadratic equations | Errors | Constraints | Error analysis | Inequalities | Convexity | Computer Science

Journal Article

Journal of global optimization, ISSN 1573-2916, 2010, Volume 50, Issue 1, pp. 39 - 57

In this paper we reformulate the generalized Nash equilibrium problem (GNEP) as a nonsmooth Nash equilibrium problem by means of a partial penalization of the difficult coupling constraints...

Nash equilibrium problem | Exact penalty function | Operations Research/Decision Theory | Jointly convex problem | Computer Science, general | Partial penalization | Generalized Nash equilibrium problem | Optimization | Economics / Management Science | Real Functions | MATHEMATICS, APPLIED | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | Game theory | Studies | Equilibrium

Nash equilibrium problem | Exact penalty function | Operations Research/Decision Theory | Jointly convex problem | Computer Science, general | Partial penalization | Generalized Nash equilibrium problem | Optimization | Economics / Management Science | Real Functions | MATHEMATICS, APPLIED | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | Game theory | Studies | Equilibrium

Journal Article

Abstract and applied analysis, ISSN 1085-3375, 5/2014, Volume 2014, pp. 1 - 13

.... Some exact penalization properties for two classes of multiobjective penalty problems are established and shown to be equivalent to the calmness condition...

EXACT PENALTY | MATHEMATICS, APPLIED | CALMNESS | QUALIFICATIONS | MATHEMATICAL PROGRAMS | Algorithms | Research | Mathematical research | Mathematical optimization | Problems | Nonlinear programming | Equilibrium | Optimization | Mathematical programming | Inequality

EXACT PENALTY | MATHEMATICS, APPLIED | CALMNESS | QUALIFICATIONS | MATHEMATICAL PROGRAMS | Algorithms | Research | Mathematical research | Mathematical optimization | Problems | Nonlinear programming | Equilibrium | Optimization | Mathematical programming | Inequality

Journal Article

Mathematics of operations research, ISSN 1526-5471, 2003, Volume 28, Issue 3, pp. 533 - 552

..., we show that generalized augmented Lagrangians present a unified approach to several classes of exact penalization results. Some equivalences among exact penalization...

Generalized augmented Lagrangian | constrained program | exact penalty function | nonlinear Lagrangian | duality | Penalty function | Sufficient conditions | Optimal solutions | Mathematical independent variables | Mathematical functions | Mathematical duality | Mathematics | Lagrangian function | Parameterization | Constrained optimization | Nonlinear Lagrangian | Duality | Exact penalty function | Constrained program | generalized augmented Lagrangian | MATHEMATICS, APPLIED | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | PENALTY | CONSTRAINED OPTIMIZATION | BARRIER METHODS | Analysis | Management science | Studies | Operations research | Nonlinear programming

Generalized augmented Lagrangian | constrained program | exact penalty function | nonlinear Lagrangian | duality | Penalty function | Sufficient conditions | Optimal solutions | Mathematical independent variables | Mathematical functions | Mathematical duality | Mathematics | Lagrangian function | Parameterization | Constrained optimization | Nonlinear Lagrangian | Duality | Exact penalty function | Constrained program | generalized augmented Lagrangian | MATHEMATICS, APPLIED | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | PENALTY | CONSTRAINED OPTIMIZATION | BARRIER METHODS | Analysis | Management science | Studies | Operations research | Nonlinear programming

Journal Article

应用数学和力学：英文版, ISSN 0253-4827, 2015, Volume 36, Issue 4, pp. 541 - 556

.... The conditions for exactness of the penalization for the exact minimax penalty function method are established by assuming that the functions constituting the considered con- strained optimization...

约束优化问题 | 拉格朗日乘数 | 极小 | 罚函数方法 | 非凸 | 编程 | 罚函数法 | 惩罚 | incave function | O242 | Mathematics | 90C26 | 49M30 | 90C30 | exactness of penalization of exact minimax penalty function | minimax penalized optimization problem | Mechanics | Applications of Mathematics | Mathematical Modeling and Industrial Mathematics | exact minimax penalty function method | invex function | Minimax penalized optimization problem | Exact minimax penalty function method | Incave function | Exactness of penalization of exact minimax penalty function | Invex function | MATHEMATICS, APPLIED | MECHANICS | SUFFICIENCY | Algorithms | Research | Mathematical research | Mathematical optimization

约束优化问题 | 拉格朗日乘数 | 极小 | 罚函数方法 | 非凸 | 编程 | 罚函数法 | 惩罚 | incave function | O242 | Mathematics | 90C26 | 49M30 | 90C30 | exactness of penalization of exact minimax penalty function | minimax penalized optimization problem | Mechanics | Applications of Mathematics | Mathematical Modeling and Industrial Mathematics | exact minimax penalty function method | invex function | Minimax penalized optimization problem | Exact minimax penalty function method | Incave function | Exactness of penalization of exact minimax penalty function | Invex function | MATHEMATICS, APPLIED | MECHANICS | SUFFICIENCY | Algorithms | Research | Mathematical research | Mathematical optimization

Journal Article

Journal of Optimization Theory and Applications, ISSN 0022-3239, 1/2009, Volume 140, Issue 1, pp. 171 - 188

This paper aims to establish duality and exact penalization results for the primal problem of minimizing an extended real-valued function in a reflexive Banach space in terms of a valley-at-0...

Valley-at-0 augmented Lagrangian function | Reflexive Banach space | Calculus of Variations and Optimal Control; Optimization | Exact penalty function | Operations Research/Decision Theory | Mathematics | Theory of Computation | Engineering, general | Applications of Mathematics | Optimization | Zero duality gap | MATHEMATICS, APPLIED | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | Mathematical optimization | Universities and colleges

Valley-at-0 augmented Lagrangian function | Reflexive Banach space | Calculus of Variations and Optimal Control; Optimization | Exact penalty function | Operations Research/Decision Theory | Mathematics | Theory of Computation | Engineering, general | Applications of Mathematics | Optimization | Zero duality gap | MATHEMATICS, APPLIED | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | Mathematical optimization | Universities and colleges

Journal Article

SIAM Journal on Control and Optimization, ISSN 0363-0129, 11/1998, Volume 37, Issue 2, pp. 617 - 652

We study theoretical and computational aspects of an exact penalization approach to mathematical programs with equilibrium constraints (MPECs...

trust region | MATHEMATICS, APPLIED | l penalty function | error bound | exact penalization | NORMAL MANIFOLDS | SENSITIVITY | SMOOTH | bilevel program | mathematical program with equilibrium constraints (MPEC) | TRUST REGION ALGORITHM | VARIATIONAL INEQUALITY CONSTRAINTS | piecewise smooth | MAPS | NONSMOOTH OPTIMIZATION | CONVERGENCE | sequential quadratic programming | Mangasarian-Fromovitz constraint qualification | AUTOMATION & CONTROL SYSTEMS

trust region | MATHEMATICS, APPLIED | l penalty function | error bound | exact penalization | NORMAL MANIFOLDS | SENSITIVITY | SMOOTH | bilevel program | mathematical program with equilibrium constraints (MPEC) | TRUST REGION ALGORITHM | VARIATIONAL INEQUALITY CONSTRAINTS | piecewise smooth | MAPS | NONSMOOTH OPTIMIZATION | CONVERGENCE | sequential quadratic programming | Mangasarian-Fromovitz constraint qualification | AUTOMATION & CONTROL SYSTEMS

Journal Article

13.
Full Text
Descent and Penalization Techniques for Equilibrium Problems with Nonlinear Constraints

Journal of Optimization Theory and Applications, ISSN 0022-3239, 2015, Volume 164, Issue 3, pp. 804 - 818

.... They are both based on the minimization of a suitable exact penalty function, but they use different rules for updating the penalization parameter and they rely on different types of line search...

Gap function | Exact penalization | Equilibria | Constraint linearization | MATHEMATICS, APPLIED | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | MODELS | ALGORITHM | OPTIMIZATION | Algorithms | Studies | Operations research | Mathematical models | Equilibrium | Approximation | Searching | Mathematical analysis | Nonlinearity | Descent | Standards | Optimization | Convergence

Gap function | Exact penalization | Equilibria | Constraint linearization | MATHEMATICS, APPLIED | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | MODELS | ALGORITHM | OPTIMIZATION | Algorithms | Studies | Operations research | Mathematical models | Equilibrium | Approximation | Searching | Mathematical analysis | Nonlinearity | Descent | Standards | Optimization | Convergence

Journal Article

IEEE Transactions on Automatic Control, ISSN 0018-9286, 04/2010, Volume 55, Issue 4, pp. 1018 - 1024

This technical note deals with robust state estimation when parametric uncertainties nonlinearly affect a plant state-space model, based on a simultaneous...

Linear systems | Estimation error | Uncertainty | Recursive state estimation | structured parametric uncertainty | Covariance matrix | Computational complexity | Filters | regularized least-squares | Numerical simulation | Robustness | State estimation | Recursive estimation | Regularized least-squares | Structured parametric uncertainty | robustness | AUTOMATION & CONTROL SYSTEMS | ENGINEERING, ELECTRICAL & ELECTRONIC | Usage | Analysis | Recursive functions | Least squares | Computer simulation | Asymptotic properties | Exact solutions | Mathematical models | Estimators | Optimization

Linear systems | Estimation error | Uncertainty | Recursive state estimation | structured parametric uncertainty | Covariance matrix | Computational complexity | Filters | regularized least-squares | Numerical simulation | Robustness | State estimation | Recursive estimation | Regularized least-squares | Structured parametric uncertainty | robustness | AUTOMATION & CONTROL SYSTEMS | ENGINEERING, ELECTRICAL & ELECTRONIC | Usage | Analysis | Recursive functions | Least squares | Computer simulation | Asymptotic properties | Exact solutions | Mathematical models | Estimators | Optimization

Journal Article

Optimal control applications & methods, ISSN 1099-1514, 2020, Volume 41, Issue 3, pp. 898 - 947

Summary The second part of our study is devoted to an analysis of the exactness of penalty functions for optimal control problems with terminal and pointwise...

optimal control | state constraint | terminal constraint | exact penalty function | fixed‐endpoint problem | MATHEMATICS, APPLIED | UNIFYING THEORY | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | FUNCTION ALGORITHM | METRIC REGULARITY | fixed-endpoint problem | AUTOMATION & CONTROL SYSTEMS | Penalty function | Stability | Equivalence | Lagrange multipliers | Mathematical analysis | Linear evolution equations | Optimal control | Nonlinear systems

optimal control | state constraint | terminal constraint | exact penalty function | fixed‐endpoint problem | MATHEMATICS, APPLIED | UNIFYING THEORY | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | FUNCTION ALGORITHM | METRIC REGULARITY | fixed-endpoint problem | AUTOMATION & CONTROL SYSTEMS | Penalty function | Stability | Equivalence | Lagrange multipliers | Mathematical analysis | Linear evolution equations | Optimal control | Nonlinear systems

Journal Article

SIAM journal on control and optimization, ISSN 1095-7138, 1991, Volume 29, Issue 4, pp. 968 - 998

... (1967), pp. 344-358] introduce a notion of exact penalization for use in the development of algorithms for constrained optimization...

MATHEMATICS, APPLIED | CONVEX PROGRAMMING | SUBDIFFERENTIAL CALCULUS | CALMNESS | MULTIFUNCTIONS | SUFFICIENT CONDITIONS | STABILITY | ALGORITHMS | 2ND ORDER | LIPSCHITZIAN | EXACT PENALTY FUNCTIONS | CONSTRAINT QUALIFICATION | LOCAL MINIMUM | EXACT PENALTY-FUNCTIONS | ORDER CONDITIONS | OPTIMALITY CONDITIONS | AUTOMATION & CONTROL SYSTEMS

MATHEMATICS, APPLIED | CONVEX PROGRAMMING | SUBDIFFERENTIAL CALCULUS | CALMNESS | MULTIFUNCTIONS | SUFFICIENT CONDITIONS | STABILITY | ALGORITHMS | 2ND ORDER | LIPSCHITZIAN | EXACT PENALTY FUNCTIONS | CONSTRAINT QUALIFICATION | LOCAL MINIMUM | EXACT PENALTY-FUNCTIONS | ORDER CONDITIONS | OPTIMALITY CONDITIONS | AUTOMATION & CONTROL SYSTEMS

Journal Article

Set-Valued and Variational Analysis, ISSN 1877-0533, 12/2016, Volume 24, Issue 4, pp. 619 - 635

In this paper we study local sharp minima of the nonlinear programming problem via exact penalization...

Local sharp minimum | 49K99 | Subderivative | 49J53 | Analysis | Probability Theory and Stochastic Processes | 65K10 | Mathematics | Smallest penalty parameter | Exact penalization | Regular subdifferential | MATHEMATICS, APPLIED | PROGRAMMING PROBLEMS | SUFFICIENT CONDITIONS | CONVERGENCE | DUALITY | CONSTRAINED OPTIMIZATION | OPTIMALITY CONDITIONS

Local sharp minimum | 49K99 | Subderivative | 49J53 | Analysis | Probability Theory and Stochastic Processes | 65K10 | Mathematics | Smallest penalty parameter | Exact penalization | Regular subdifferential | MATHEMATICS, APPLIED | PROGRAMMING PROBLEMS | SUFFICIENT CONDITIONS | CONVERGENCE | DUALITY | CONSTRAINED OPTIMIZATION | OPTIMALITY CONDITIONS

Journal Article

Optimization, ISSN 1029-4945, 2015, Volume 65, Issue 6, pp. 1167 - 1202

In this article, we develop a theory of exact linear penalty functions that generalizes and unifies most of the results on exact penalization existing in the literature...

Penalty function | calmness | error bounds | Palais-Smale condition | perturbation function | exact penalization | Palais–Smale condition | Studies | Fines & penalties | Optimization | Mathematics - Optimization and Control

Penalty function | calmness | error bounds | Palais-Smale condition | perturbation function | exact penalization | Palais–Smale condition | Studies | Fines & penalties | Optimization | Mathematics - Optimization and Control

Journal Article