Incremental and Total Optimization of Separable Functionals with Constraints
Lawrence D. Stone · SIAM Journal on Control and Optimization · 1976
Functionals E (real-valued) and C (vector-valued) are defined by $E(q) = \int_X {e(x,q(x))\mu (dx)} $ and $C(q) = \int_X {c(x,q(x))\mu (dx)} $, where $\mu $ is a Borel regular, nonatomic measure defined on a Borel subset X of a complete separable metric space. Let $\omega $ be the positive integers. Let $q_0 ,q_1 , \cdots $, be extended real functions such that $q_0 = - \infty $ and $q_i \geqq q_{i - 1} $ for $i \in \omega $. A function $q^ * $ is called optimal if $E(q^ * ) = \max \{ E(q):C(q) = C(q^ * )\} $ The sequence $(q_1 ,q_2 , \cdots )$ is incrementally optimal if $E(q_i ) = \max \{ E(p):p \geqq q_{i - 1} {\text{ and }}C(p) = C(q_i )\} $ for $i \in \omega $ and totally optimal if $q_i $ is optimal for $i \in \omega $. Under appropriate measurability assumptions, it is shown that if $c(x, \cdot )$ is real-valued and increasing for $x \in X$, then an incrementally optimal sequence such that $| {E(q_i )} | < \infty $ and $C(q_i ) \in $ interior range C for $i \in \omega $ is totally optimal. A counterexample is given to show that an extension of this result to multiple constraints fails even if $e(x, \cdot )$ and $c(x, \cdot )$ are linear for $x \in X$. In the case of a single constraint, the existence of optimal functions is proved under conditions which allow the range of C to be unbounded above.