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

我们设计，分析和测试了用于解决结构化凸优化问题的黄金比率原始对偶算法（GRPDA），其中目标函数是两个封闭的固有凸函数的和，其中一个涉及线性变换的组合。GRPDA保留了经典原始对偶算法（PDA）的所有有利功能，即原始变量和对偶变量以Gauss-Seidel方式进行更新，并且每次迭代的成本主要取决于评估的近点映射。两个分量函数和两个矩阵向量乘法。与经典PDA（需要外推的步骤）相比，GRPDA的新颖之处在于它是基于基本上整个迭代轨迹的凸组合构造的。我们证明，只要凸组合参数的倒数在上面由黄金分割率限定，GRPDA就能在比经典PDA更大的参数范围内收敛，从而说明了算法的名称。一个\（\ mathcal {O}（1 / N）\）遍历收敛速度结果也是基于原始对偶间隙函数建立的，其中N表示迭代次数。当原始问题或对偶问题中的任何一个是强凸时，都构造了一个加速GRPDA，以将遍历收敛速度从\（\ mathcal {O}（1 / N）\）提高到\（\ mathcal {O}（1 / N ^ 2）\）。此外，对于正则化最小二乘和线性等式约束问题，我们表明凸组合参数的倒数可以从黄金比例扩展到2，同时可以采取松弛步骤。我们对LASSO，非负最小二乘和minimax矩阵博弈问题的初步数值结果，与一些最新的相关算法进行了比较，证明了所提出算法的有效性。