Abstract
In this paper, we proposed a logarithmic-quadratic proximal alternating direction method for structured variational inequalities. The new iterate is obtained by a convex combination of the previous point and the one generated by a projection-type method along a new descent direction. Global convergence of the new method is proved under certain assumptions. We also reported some numerical results to illustrate the efficiency of the proposed method.MSC: 90C33, 49J40.
Highlights
The problem we are concerned with in this paper is for the following variational inequalities: find u ∈ such that u – u T F(u) ≥, ∀u ∈, ( . ) with x f (x) u =, F(u) = and y g(y)= (x, y) | x ∈ Rn++, y ∈ Rm++, Ax + By = b, where A ∈ Rl×n, B ∈ Rl×m are given matrices, b ∈ Rl is a given vector, and f : Rn++ → Rn, g : Rm++ → Rm are given monotone operators
Li [ ] presented an logarithmic-quadratic proximal (LQP)-based prediction-correction method, the new iterate is obtained by a convex combination of the previous point and the one generated by a projection-type method along this descent direction
In the present paper, inspired by the above cited works and by the recent works going in this direction, we proposed a new LQP-based prediction-correction method; the new iterate is obtained by a convex combination of the previous point and the one generated by a projection-type method along another descent direction
Summary
For given (xk, yk, λk) ∈ Rn++ × Rm++ × Rl, the new iterative (xk+ , yk+ , λk+ ) is obtained via the following steps. Li [ ] presented an LQP-based prediction-correction method, the new iterate is obtained by a convex combination of the previous point and the one generated by a projection-type method along this descent direction. Prediction step: For a given wk = (xk, yk, λk) ∈ Rn++ × Rm++ × Rl, and μ ∈ ( , ), the predictor wk = (xk, yk, λk) ∈ Rn++ × Rm++ × Rl is obtained via solving the following system: f (x) – AT λk – H Ax + Byk – b + R x – xk + μ xk – Xk x– = , g(y) – BT λk – H(Ax + By – b) + S y – yk + μ yk – Yk y– = , λk = λk – H Axk + Byk – b .
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