A Regularized Smoothing Newton Method for Box Constrained Variational Inequality Problems with P0-Functions

Qi, Houduo

doi:10.1137/s1052623497324047

Cited by 84 publications

(53 citation statements)

References 23 publications

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“…This proves (43). If H is strongly semismooth at z * , then from the above argument we can easily get (45).…”

Section: Superlinear and Quadratic Convergencementioning

confidence: 63%

“…Moreover, we require F to be a P 0 -function on n + only instead of on n . After the announcement of this paper, our method has soon been used to solve various regularized smoothing equations to get stronger global convergent results [53,43,60] and to solve extended vertical linear complementarity problems [44].…”

Section: Discussionmentioning

confidence: 99%

“…Suppose that H is semismooth at z * and that BVI is R-regular at x * . Then (42) and (43) in Theorem 8 hold. Furthermore, if H is strongly semismooth at z * , then (44) and (45) in Theorem 8 hold.…”

Section: Theorem 9 Suppose That Assumptions 1 and 2 Are Satisfied And Zmentioning

confidence: 90%

“…Further, if the R-regularity holds at x * , then the whole sequence {z k } converges to z * , and (42) and (43) (44) and (45) in Theorem 8 hold. (18) or (20).…”

Section: Is a P-matrix; (Ii) If φ(·) Is Defined By Either (16) Or (18mentioning

confidence: 95%

See 3 more Smart Citations

A new look at smoothing Newton methods for nonlinear complementarity problems and box constrained variational inequalities

Qi¹,

Sun²,

Zhou³

2000

Math. Program.

347

226

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Abstract. In this paper we take a new look at smoothing Newton methods for solving the nonlinear complementarity problem (NCP) and the box constrained variational inequalities (BVI). Instead of using an infinite sequence of smoothing approximation functions, we use a single smoothing approximation function and Robinson's normal equation to reformulate NCP and BVI as an equivalent nonsmooth equation H(u, x) = 0, where H : 2n → 2n , u ∈ n is a parameter variable and x ∈ n is the original variable. The central idea of our smoothing Newton methods is that we construct a sequence {z k = (u k , x k )} such that the mapping H(·) is continuously differentiable at each z k and may be non-differentiable at the limiting point of {z k }. We prove that three most often used Gabriel-Moré smoothing functions can generate strongly semismooth functions, which play a fundamental role in establishing superlinear and quadratic convergence of our new smoothing Newton methods. We do not require any function value of F or its derivative value outside the feasible region while at each step we only solve a linear system of equations and if we choose a certain smoothing function only a reduced form needs to be solved. Preliminary numerical results show that the proposed methods for particularly chosen smoothing functions are very promising.

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“…This proves (43). If H is strongly semismooth at z * , then from the above argument we can easily get (45).…”

Section: Superlinear and Quadratic Convergencementioning

confidence: 63%

Section: Discussionmentioning

confidence: 99%

Section: Theorem 9 Suppose That Assumptions 1 and 2 Are Satisfied And Zmentioning

confidence: 90%

“…Further, if the R-regularity holds at x * , then the whole sequence {z k } converges to z * , and (42) and (43) (44) and (45) in Theorem 8 hold. (18) or (20).…”

Section: Is a P-matrix; (Ii) If φ(·) Is Defined By Either (16) Or (18mentioning

confidence: 95%

See 2 more Smart Citations

A new look at smoothing Newton methods for nonlinear complementarity problems and box constrained variational inequalities

Qi¹,

Sun²,

Zhou³

2000

Math. Program.

347

226

View full text Add to dashboard Cite

show abstract

“…We omit the details here. [7,13,24,33]. It is known that Assumption 3.1 is weaker than those required by most existing smoothing (non-interior continuation) Newton-type methods [12].…”

Section: Algorithm 31 (A Smoothing Newton-type Algorithm)mentioning

confidence: 99%

A Smoothing Newton-Type Algorithm of Stronger Convergence for the Quadratically Constrained Convex Quadratic Programming

Huang

Sun

Zhao

2006

Comput Optim Applic

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A novel multidimensional penalty‐free approach for constrained optimal control of switched control systems

Sun

Wang

et al. 2020

Intl J Robust & Nonlinear

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In this article, a novel multidimensional penalty-free method is proposed for the optimal control problem of switched control systems with path constraints. First, by introducing a Boolean vector, the switched control system and path constraints are transformed into affine forms. Typically, the complementary constrained method and piecewise constant function are used to relax and approximate the Boolean vector. Moreover, numerical methods are used to solve the approximated constrained optimal control problem (COCP), which relies on a discrete grid of the control function by using a piecewise linear function. Furthermore, the approximated COCP of affine control systems is equivalently transformed into a constrained dynamic parameter optimization problem by a refined time-scaling transformation. Finally, the global convergence of the proposed multidimensional penalty-free algorithm is proved and the feasibility of the algorithm is verified by numerical examples.

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A Regularized Smoothing Newton Method for Box Constrained Variational Inequality Problems with P0-Functions

Cited by 84 publications

References 23 publications

A new look at smoothing Newton methods for nonlinear complementarity problems and box constrained variational inequalities

A new look at smoothing Newton methods for nonlinear complementarity problems and box constrained variational inequalities

A Smoothing Newton-Type Algorithm of Stronger Convergence for the Quadratically Constrained Convex Quadratic Programming

A novel multidimensional penalty‐free approach for constrained optimal control of switched control systems

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