A family of operator splitting methods revisited. (English) Zbl 1201.65110
Let \(H\) be a real Hilbert spacee; and \(A,B\) be (multivalued) maximal monotone operators from \(H\) to itself. The variable scaling DPRV algorithm Step 0: \(x^0\in H\), \(\mu_0\in (0,\infty)\), \(k:=0\);
Step 1: \(a^k\in A(x^k), (y^k,b^k)\in B\): \(y^k+\mu_kb^k=x^k-\mu_ka^k\), \(x^k=y^k\) \(\Rightarrow\) stop;
Step 2: \(\gamma_k> 0\), \((x^{k+1},a^{k+1})\in A\): \(x^{k+1}+\mu_ka^{k+1}=x^k+\mu_ka^k-\gamma_k(x^k-y^k)\);
Step 3: \(\tau_k\in [0,1)\), \(\mu_{k+1}\in [(1-\tau_k)\mu_k,(1+\tau_k)\mu_k]\), \(k:=k+1\), go to (Step 1)
for solving the inclusion \(0\in A(x)+B(x)\) is extended to the case when the parameters may vary in a non-monotonical way from iteration to iteration. Sufficient conditions are given so as to get a linear rate of convergence.
Step 1: \(a^k\in A(x^k), (y^k,b^k)\in B\): \(y^k+\mu_kb^k=x^k-\mu_ka^k\), \(x^k=y^k\) \(\Rightarrow\) stop;
Step 2: \(\gamma_k> 0\), \((x^{k+1},a^{k+1})\in A\): \(x^{k+1}+\mu_ka^{k+1}=x^k+\mu_ka^k-\gamma_k(x^k-y^k)\);
Step 3: \(\tau_k\in [0,1)\), \(\mu_{k+1}\in [(1-\tau_k)\mu_k,(1+\tau_k)\mu_k]\), \(k:=k+1\), go to (Step 1)
for solving the inclusion \(0\in A(x)+B(x)\) is extended to the case when the parameters may vary in a non-monotonical way from iteration to iteration. Sufficient conditions are given so as to get a linear rate of convergence.
Reviewer: Mihai Turinici (Iaşi)
MSC:
| 65K10 | Numerical optimization and variational techniques |
| 46N10 | Applications of functional analysis in optimization, convex analysis, mathematical programming, economics |
| 49M30 | Other numerical methods in calculus of variations (MSC2010) |
| 47J22 | Variational and other types of inclusions |
| 49J21 | Existence theories for optimal control problems involving relations other than differential equations |
| 47H05 | Monotone operators and generalizations |
| 90C33 | Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming) |
Keywords:
splitting; convergence; nonlinear complementarity problem; quadratic minimization; Hilbert space; maximal monotone operators; variable scaling DPRV algorithm; variable scaling DPRV algorithm; inclusionReferences:
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