  Using convex combination and linesearch techniques, we introduce a novel
primal-dual algorithm for solving structured convex-concave saddle point
problems with a generic smooth nonbilinear coupling term. Our adaptive
linesearch strategy works under specific local smoothness conditions, allowing
for potentially larger stepsizes. For an important class of structured convex
optimization problems, the proposed algorithm reduces to a fully adaptive
proximal gradient algorithm without linesearch, thereby representing an
advancement over the golden ratio algorithm delineated in [Y. Malitsky, Math.
Program. 2020]. We establish global pointwise and ergodic sublinear convergence
rate of the algorithm measured by the primal-dual gap function in the general
case. When the coupling term is linear in the dual variable, we measure the
convergence rate by function value residual and constraint violation of an
equivalent constrained optimization problem. Furthermore, an accelerated
algorithm achieving the faster O(1/N^2) ergodic convergence rate is presented
for the strongly convex case, where N denotes the iteration number. Our
numerical experiments on quadratically constrained quadratic programming and
sparse logistic regression problems indicate the new algorithm is significantly
faster than the comparison algorithms.
