An Inexact Uzawa Algorithmic Framework for Nonlinear Saddle Point Problems with Applications to Elliptic Optimal Control Problem


We consider a class of nonlinear saddle point problems with various applications in PDEs and optimal control problems and propose an algorithmic framework based on some inexact Uzawa methods in the literature. Under mild conditions, the convergence of this algorithmic framework is uniformly proved and the linear convergence rate is estimated. We take an elliptic optimal control problem with control constraints as an example to illustrate how to choose application-tailored preconditioners to generate specific and efficient algorithms by the algorithmic framework. The resulting algorithm does not need to solve any optimization subproblems or systems of linear equations in its iteration; each of its iterations only requires the projection onto a simple admissible set, four algebraic multigrid V-cycles, and a few matrix-vector multiplications. Its numerical efficiency is then demonstrated by some preliminary numerical results.


  1. nonlinear saddle point problems
  2. inexact Uzawa method
  3. convergence analysis
  4. subregularity
  5. linear convergence rate
  6. elliptic optimal control problem

MSC codes

  1. 49J20
  2. 65M12
  3. 90C25

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Information & Authors


Published In

cover image SIAM Journal on Numerical Analysis
SIAM Journal on Numerical Analysis
Pages: 2656 - 2684
ISSN (online): 1095-7170


Submitted: 22 February 2019
Accepted: 25 September 2019
Published online: 12 November 2019


  1. nonlinear saddle point problems
  2. inexact Uzawa method
  3. convergence analysis
  4. subregularity
  5. linear convergence rate
  6. elliptic optimal control problem

MSC codes

  1. 49J20
  2. 65M12
  3. 90C25



Funding Information

Hong Kong Research Grants Council : 12302318

Funding Information

University of Hong Kong : 201807159005

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