Cyclic exact block minimization

Let $d,n\ge 1$, let $B_1,\dots ,B_n$ partition the coordinates of ${\mathbb{R}}^d$ into nonempty blocks, let $F:{\mathbb{R}}^d\to (-\mathrm{\infty },+\mathrm{\infty }]$, and let $x^0$ satisfy $F(x^0)<+\mathrm{\infty }$. A cyclic exact block-minimization run with sweep budget $K\in \mathbb{N}$ processes blocks in the fixed order $1,\dots ,n$ for at most $K$ sweeps. At the update of block $i$, hold all other coordinates at their current values and form $M_i(x_{-i})=\arg {\min }_{u\in {\mathbb{R}}^{\mathrm{\mid }{B}_i\mathrm{\mid }}}F(x_{-i},u)\mathrm{.}$ If $M_i(x_{-i})=\mathrm{\varnothing }$, return the last iterate and status block-minimum-failure. Otherwise replace the current block by any member of $M_i(x_{-i})$. If a minimizer is nonunique, the procedure is relation-valued and every member is an admissible choice; the convergence theorem on this page assumes unique continuous block-minimizer maps. After $K$ complete sweeps, return the current iterate and status budget-limit. The finite budget does not certify a stationary point.