Relaxation methods with step regulation for solving constrained optimization problems (Q1915675)

We study the problem \[ \min_{x\in D} f(x),\tag{1} \] where \(f(x)\) is a continuously differentiable pseudoconvex function satisfying Condition A (introduced in [\textit{Z. R. Gabidulinna}, J. Sov. Math. 50, No. 5, 1803-1809 (1990); translation from Issled. Prikl. Mat. 14, 15-25 (1987; Zbl 0706.90061)]): A continuous function \(f\) satisfies Condition A on the set \(D\) if there exists \(\kappa>0\) such that \[ f(\alpha x+(1- \alpha)y)\geq \alpha f(x)+ (1-\alpha) f(y)- \alpha(1- \alpha)\kappa\| x-y\|^2\;\forall x,y\in D,\;\forall\alpha\in [0,1]. \] We propose a relaxation method for solving the problem (1). Depending on the method of choosing the size of the recursion step, one can obtain various versions of the algorithm.