A Golden Ratio Primal–Dual Algorithm for Structured Convex Optimization

We design, analyze and test a golden ratio primal–dual algorithm (GRPDA) for solving structured convex optimization problem, where the objective function is sum of two closed proper functions, one which involves composition with linear transform. GRPDA preserves all favorable features classical (PDA), i.e., primal dual variables are updated in Gauss–Seidel manner, per iteration cost dominated by evaluation proximal point mappings component functions matrix-vector multiplications. Compared PDA, takes an extrapolation step, novelty that it constructed based on combination essentially whole trajectory. show converges within broader range parameters than provided reciprocal parameter bounded above ratio, explains name algorithm. An $$\mathcal {O}(1/N)$$ ergodic convergence rate result also established gap function, N denotes number iterations. When either or problem strongly convex, accelerated to improve from {O}(1/N^2)$$ . Moreover, we regularized least-squares equality constrained problems can be extended 2 meanwhile relaxation step taken. Our preliminary numerical results LASSO, nonnegative minimax matrix game problems, comparisons some state-of-the-art relative algorithms, demonstrate efficiency proposed algorithms.