A new smoothing method for solving nonlinear complementarity problems

Zhu, Jianguang; Hao, Bin

doi:10.1515/math-2019-0011

Cited by 13 publications

(9 citation statements)

References 49 publications

(56 reference statements)

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“…The first, a nonsmooth quasi-newton method proposed in [5] which we call Algorithm 2. The second, a smoothing Jacobian method proposed in [18] which, unlike our proposal, uses a smoothing of the Fischer function (Algorithm 1 with λ = 2), which Algorithm 3 and, the third, a smooth Newton method proposed recently in [32], which we call Algorithm 4. We vary λ in two forms obtaining two versions of our algorithm, namely, Method 1: we use the dynamic choice of λ used in [5], (this strategy combines the efficiency of Fisher function far from the solution with that of the minimum function near to it), Method 2: we vary randomly λ in the interval (0, 4) .…”

Section: Numerical Resultsmentioning

confidence: 99%

“…For the numerical test, we consider nine complementarity problems associated with the functions Kojima-Shindo (Koj-Shi), Kojima-Josephy (Koj-Jo), Mathiesen modificado (Math mod), Mathiesen (Mathiesen) Billups (Billups) [7], [25]; Nash-Cournot (Nash-Co) [16], Hock-Schittkowski (HH 66 ) [32], Geiger-Kanzow (Geiger-Kanzow) [15], Ahn (Ahn) [2]. We implemented Algorithms 1 (with Methods 1 and 2) and the test functions in MATLAB and use the following starting points taking from [5], [32],…”

Section: Numerical Resultsmentioning

confidence: 99%

“…We developed their convergence theory and did numerical tests to analyze their global and local performance. Our proposal presents some advantages in terms of global convergence compared to other methods as those proposed in [5], [18], and [32].…”

Section: Final Remarksmentioning

confidence: 99%

See 2 more Smart Citations

Un algoritmo global con jacobiano suavizado para problemas de complementariedad no lineal

Sánchez¹,

Perez²,

Martínez³

2021

Rev Integr temas mat

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In this paper, we use the smoothing Jacobian strategy to propose a new algorithm for solving complementarity problems based on its reformulation as a nonsmooth system of equations. This algorithm can be seen as a generalization of the one proposed in [18]. We develop its global convergence theory and under certain assumptions, we demonstrate that the proposed algorithm converges locally and, q-superlinearly or q-quadratically to a solution of the problem. Some numerical experiments show a good performance of this algorithm.

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Section: Numerical Resultsmentioning

confidence: 99%

Section: Numerical Resultsmentioning

confidence: 99%

See 1 more Smart Citation

Un algoritmo global con jacobiano suavizado para problemas de complementariedad no lineal

Sánchez¹,

Perez²,

Martínez³

2021

Rev Integr temas mat

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show abstract

“…This results in the fact that the reconstruction by using TGV regularization can preserve edges while suppressing staircase effect. In order to solve the TGV model efficiently, many optimization algorithms have been proposed, such as Newton's method, split Bregman method, alternating direction method of multipliers, and gradient descent method [35][36][37][38][39][40][41][42][43]. Experiments show that TGV has the superior performance to TV based regularization models in image reconstruction.…”

Section: Introductionmentioning

confidence: 99%

Image Restoration by Second‐Order Total Generalized Variation and Wavelet Frame Regularization

Zhu

Hao

2019

Complexity

Self Cite

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It has been proved that total generalized variation (TGV) can better preserve edges while suppressing staircase effect. In this paper, we propose an effective hybrid regularization model based on second-order TGV and wavelet frame. The proposed model inherits the advantages of TGV regularization and wavelet frame regularization, can eliminate staircase effect while protecting the sharp edge, and simultaneously has good capability of sparsely estimating the piecewise smooth functions. The alternative direction method of multiplier (ADMM) is employed to solve the new model. Numerical results show that our proposed model can preserve more details and get higher image visual quality than some current state-of-the-art methods.

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“…On the other hand, the investigation of the regularity of discrete maximal operators has also attracted the attention of many authors (cf. [2,5,7,18,21,24,27,29,32,36,37]). Let us recall some definitions and background.…”

Section: Introductionmentioning

confidence: 99%

Endpoint regularity of discrete multilinear fractional nontangential maximal functions

Zhang

2019

Adv Differ Equ

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Given $m\geq 1$ m ≥ 1 , $0\leq \lambda \leq 1$ 0 ≤ λ ≤ 1 , and a discrete vector-valued function $\vec{f}=(f_{1},\ldots,f_{m})$ f → = ( f 1 , … , f m ) with each $f_{j}:\mathbb{Z} ^{d}\rightarrow \mathbb{R}$ f j : Z d → R , we consider the discrete multilinear fractional nontangential maximal operator $$ \mathrm{M}_{\alpha,\mathcal{B}}^{\lambda }(\vec{f}) (\vec{n})=\mathop{\sup_{r>0, \vec{x}\in \mathbb{R}^{d}}}_{ \vert \vec{n}-\vec{x} \vert \leq \lambda r}\frac{1}{N(B _{r}(\vec{x}))^{m-\frac{\alpha }{d}}} \prod _{j=1}^{m}\sum_{\vec{k}\in B_{r}(\vec{x})\cap \mathbb{Z}^{d}} \bigl\vert f_{j}(\vec{k}) \bigr\vert , $$ M α , B λ ( f → ) ( n → ) = sup r > 0 , x → ∈ R d | n → − x → | ≤ λ r 1 N ( B r ( x → ) ) m − α d ∏ j = 1 m ∑ k → ∈ B r ( x → ) ∩ Z d | f j ( k → ) | , where $\mathcal{B}$ B is the collection of all open balls $B\subset \mathbb{R}^{d}$ B ⊂ R d , $B_{r}(\vec{x})$ B r ( x → ) is the open ball in $\mathbb{R}^{d}$ R d centered at $\vec{x}\in \mathbb{R}^{d}$ x → ∈ R d with radius r, and $N(B_{r}(\vec{x}))$ N ( B r ( x → ) ) is the number of lattice points in the set $B_{r}(\vec{x})$ B r ( x → ) . We show that the operator $\vec{f}\mapsto |\nabla \mathrm{M}_{\alpha, \mathcal{B}}^{\lambda }(\vec{f})|$ f → ↦ | ∇ M α , B λ ( f → ) | is bounded and continuous from $\ell ^{1}(\mathbb{Z}^{d})\times \ell ^{1}(\mathbb{Z} ^{d})\times \cdots \times \ell ^{1}(\mathbb{Z}^{d})$ ℓ 1 ( Z d ) × ℓ 1 ( Z d ) × ⋯ × ℓ 1 ( Z d ) to $\ell ^{q}(\mathbb{Z} ^{d})$ ℓ q ( Z d ) if $0\leq \alpha < md$ 0 ≤ α < m d and $q\geq 1$ q ≥ 1 such that $q>\frac{d}{md- \alpha +1}$ q > d m d − α + 1 . We also prove that the same result also holds for the discrete multilinear fractional nontangential maximal operators associated with cubes. These results we obtained represent significant and natural extensions of what was known previously.

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A new smoothing method for solving nonlinear complementarity problems

Cited by 13 publications

References 49 publications

Un algoritmo global con jacobiano suavizado para problemas de complementariedad no lineal

Un algoritmo global con jacobiano suavizado para problemas de complementariedad no lineal

Image Restoration by Second‐Order Total Generalized Variation and Wavelet Frame Regularization

Endpoint regularity of discrete multilinear fractional nontangential maximal functions

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