  This paper tackles the challenging problem of finding global optimal
solutions for two-stage stochastic programs with continuous decision variables
and nonconvex recourse functions. We introduce a two-phase approach. The first
phase involves the construction of a polynomial lower bound for the recourse
function through a linear optimization problem over a nonnegative polynomial
cone. Given the complex structure of this cone, we employ semidefinite
relaxations with quadratic modules to facilitate our computations. In the
second phase, we solve a surrogate first-stage problem by substituting the
original recourse function with the polynomial lower approximation obtained in
the first phase. Our method is particularly advantageous for two reasons: it
not only generates global lower bounds for the nonconvex stochastic program,
aiding in the certificate of global optimality for prospective solutions like
stationary solutions computed from other methods, but it also yields an
explicit polynomial approximation for the recourse function through the
solution of a linear conic optimization problem, where the number of variables
is independent of the support of the underlying random vector. Therefore, our
approach is particularly suitable for the case where the random vector follows
a continuous distribution or when dealing with a large number of scenarios.
Numerical experiments are conducted to demonstrate the effectiveness of our
proposed approach.
