A primal-dual interior-point method capable of rapidly detecting infeasibility for nonlinear programs

Dai, Yu‐Hong; Li, Xinwei; Sun, Jie

doi:10.3934/jimo.2018190

Cited by 21 publications

(25 citation statements)

References 34 publications

(117 reference statements)

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“…As a comparison, IPOPT fails to find the solution and terminates at x * = −1.0000 in 13 iterations. In interior-point framework, this problem has been solved by the recently developed methods of [16] and [24] in totally 16 and 19 iterations, respectively. When solving the HS test problems of the CUTE collection, Algorithm 1 was terminated as either E(x k , λ k , s k ) ≤ ǫ, or the number of iterations is larger than 500 (which is the default setting of IPOPT), the step-size is too small (α k ≤ δ 40 ), the coefficient matrix of the system (3.10) is degenerate.…”

Section: Local Convergencementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…With the help of a logarithmic barrier augmented Lagrangian function, [16] proposed a biparametric primal-dual nonlinear system which corresponds to a KKT point and an infeasible stationary point of the original problem, respectively, as one of two parameters is zero. The method in [16] always generated interior-point iterates without any truncation of the step. Based on the equivalence of a positive relaxation problem to the logarithmic-barrier subproblem, Liu and Dai [24] presented a globally convergent primal-dual interior-point relaxation method for nonlinear programs, which did not require any primal or dual iterate to be an interior point.…”

Section: Introductionmentioning

confidence: 99%

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A globally convergent primal-dual interior-point relaxation method for nonlinear programs

Dai

2019

Math. Comp.

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Based on solving an equivalent parametric equality constrained mini-max problem of the classic logarithmic-barrier subproblem, we present a novel primal-dual interior-point relaxation method for nonlinear programs. In the proposed method, the barrier parameter is updated in every step as done in interior-point methods for linear programs, which is prominently different from the existing interior-point methods and the relaxation methods for nonlinear programs. Since our update for the barrier parameter is autonomous and adaptive, the method has potential of avoiding the possible difficulties caused by the unappropriate initial selection of the barrier parameter and speeding up the convergence to the solution. Moreover, it can circumvent the jamming difficulty of global convergence caused by the interior-point restriction for nonlinear programs and improve the ill conditioning of the existing primal-dual interiorpoint methods as the barrier parameter is small. Under suitable assumptions, our method is proved to be globally convergent and locally quadratically convergent. The preliminary numerical results on a well-posed problem for which many line-search interior-point methods fail to find the minimizer and a set of test problems from the CUTE collection show that our method is efficient.

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Section: Local Convergencementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A globally convergent primal-dual interior-point relaxation method for nonlinear programs

Dai

2019

Math. Comp.

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“…Burke, Curtis and Wang [5] considered the general program with equality and inequality constraints, and proved that their SQP method has strong global convergence and rapid convergence to the KKT point, and has superlinear/quadratic convergence to an infeasible stationary point. Recently, Dai, Liu and Sun [7] proposed a primal-dual interior-point method, which can be superlinearly or quadratically convergent to the Karush-Kuhn-Tucker point if the original problem is feasible, and can be superlinearly or quadratically convergent to the infeasible stationary point when the problem is infeasible.…”

Section: Introductionmentioning

confidence: 99%

Optimization with Least Constraint Violation

LiweiZhang¹

2021

CSIAM-AM

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Study about theory and algorithms for nonlinear programming usually assumes that the feasible region of the problem is nonempty. However, there are many important practical nonlinear programming problems whose feasible regions are not known to be nonempty or not, and optimizers of the objective function with the least constraint violation prefer to be found. A natural way for dealing with these problems is to extend the nonlinear programming problem as the one optimizing the objective function over the set of points with the least constraint violation. Firstly, the minimization problem with least constraint violation is proved to be an Lipschitz equality constrained optimization problem when the original problem is a convex nonlinear programming problem with possible inconsistent constraints, and it can be reformulated as an MPCC problem; And the minimization problem with least constraint violation is relaxed to an MPCC problem when the original problem is an nonlinear programming problem with possible inconsistent non-convex constraints. Secondly, for nonlinear programming problems with possible inconsistent constraints, it is proved that a local minimizer of the MPCC problem is an M-stationary point and an elegant necessary optimality condition, named as L-stationary condition, is established from the classical optimality theory of Lipschitz continuous optimization. Thirdly, properties of the penalty method for the minimization problem with the least constraint violation are developed and the proximal gradient method for the penalized problem is studied. Finally, the smoothing Fischer-Burmeister function method is constructed for solving the MPCC problem related to minimizing the objective function with the least constraint violation. It is demonstrated that, when the positive smoothing parameter approaches to zero, any point in the outer limit of the KKT-point mapping is an L-stationary point of the equivalent MPCC problem.

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“…Burke, Curtis and Wang [3] considered the general program with equality and inequality constraints, and proved that their SQP method has strong global convergence and rapid convergence to the KKT point, and has superlinear/quadratic convergence to an infeasible stationary point. Recently, Dai, Liu and Sun [10] proposed a primal-dual interior-point method, which can be superlinearly or quadratically convergent to the KKT point if the original problem is feasible, and can be superlinearly or quadratically convergent to the infeasible stationary point when the problem is infeasible.…”

Section: Introductionmentioning

confidence: 99%

The Augmented Lagrangian Method Can Approximately Solve Convex Optimization with Least Constraint Violation

Dai¹,

Zhang²

2021

Preprint

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There are many important practical optimization problems whose feasible regions are not known to be nonempty or not, and optimizers of the objective function with the least constraint violation prefer to be found. A natural way for dealing with these problems is to extend the nonlinear optimization problem as the one optimizing the objective function over the set of points with the least constraint violation. This leads to the study of the shifted problem. This paper focuses on the constrained convex optimization problem. The sufficient condition for the closedness of the set of feasible shifts is presented and the continuity properties of the optimal value function and the solution mapping for the shifted problem are studied. Properties of the conjugate dual of the shifted problem are discussed through the relations between the dual function and the optimal value function. The solvability of the dual of the optimization problem with the least constraint violation is investigated. It is shown that, if the least violated shift is in the domain of the subdifferential of the optimal value function, then this dual problem has an unbounded solution set. Under this condition, the optimality conditions for the problem with the least constraint violation are established in term of the augmented Lagrangian. It is shown that the augmented Lagrangian method has the properties that the sequence of shifts converges to the least violated shift and the sequence of multipliers is unbounded. Moreover, it is proved that the augmented Lagrangian method is able to find an approximate solution to the problem with the least constraint violation.

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A primal-dual interior-point method capable of rapidly detecting infeasibility for nonlinear programs

Cited by 21 publications

References 34 publications

A globally convergent primal-dual interior-point relaxation method for nonlinear programs

A globally convergent primal-dual interior-point relaxation method for nonlinear programs

Optimization with Least Constraint Violation

The Augmented Lagrangian Method Can Approximately Solve Convex Optimization with Least Constraint Violation

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