Consider the problem (MP) \(J(u)\to\inf\), \(u\in U\), where the function \(J(u)\) is defined, finite, convex, and lower semicontinuous on \(U\). \(U\) is a given convex closed set in a Hilbert space \(H\). The continuous proximal method for this problem involves finding the solution of the differential equation: \(\dot u(t)=\text{arg min}[{1\over 2}|z-u|^2+\beta(t)J(z)]\), \(t\geq 0\), \(z\in U\), \(u\in H\), where \(\beta(t)\) is the parameter. Consider instead of \(J(u)\) its approximation \(J_\delta(u)\) with prescribed accuracy \(\delta(t)\).
The continuous regularized proximal method solves the modified problem \((\text{MP}_\varepsilon)\), where the approximation satisfies the relation: \(|J_\delta(u)-J(u)|\leq\delta(t)(1+|u|^2)\) for \(\dot\nu(t)=pr_\varepsilon(\nu(t),t)-\nu(t)\), \(\nu(0)\geq u_0\), where \(\psi_{\nu(t),t}(pr_\varepsilon(\nu(t),t)\leq\inf\psi_{\nu(t),t}(z)+\varepsilon(t),\varepsilon(t)\geq 0\), \(\psi_{\nu,t}(z)={1\over 2}|z-\nu|^2+\beta(t)(J_\delta(z)+{1\over 2}\alpha(t)|z|^2)\), \(u\in U\).
Sufficient compatibility and convergence conditions on \(\alpha(t)\), \(\beta(t)\) and \(\varepsilon(t)\) for which the solutions of the MP\(_\varepsilon\) problem converges to the solution of the MP problem are proved in three versions: 1) Exact version by using the Tikhonov function; 2) \(J(u)\) substituted by known approximations \(J_\delta=J(u,t)\), \(\delta(t)\to 0\) if \(t\to\infty\); 3) \(J_\delta(u)\) for fixed constant \(\delta(t)\geq\delta>0\).