Math. Program., Ser. B (2018) 168:599–613
FULL LENGTH PAPER
Quadratic two-stage stochastic optimization
with coherent measures of risk
· Li-Zhi Liao
· Brian Rodrigues
Received: 11 February 2016 / Accepted: 26 February 2017 / Published online: 4 March 2017
© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2017
Abstract A new scheme to cope with two-stage stochastic optimization problems uses
a risk measure as the objective function of the recourse action, where the risk measure
is deﬁned as the worst-case expected values over a set of constrained distributions.
This paper develops an approach to deal with the case where both the ﬁrst and second
stage objective functions are convex linear-quadratic. It is shown that under a standard
set of regularity assumptions, this two-stage quadratic stochastic optimization problem
with measures of risk is equivalent to a conic optimization problem that can be solved
in polynomial time.
Keywords Conic duality · Quadratic programs · Risk measures · Stochastic
Mathematics Subject Classiﬁcation 90C15 · 90C25 · 90C34
This paper is dedicated to Terry Rockafellar in celebration of his 80th birthday.
School of Science and CIC, Curtin University, Bentley, Australia
School of Business, National University of Singapore, Singapore, Singapore
Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong
Lee Kong Chian School of Business, Singapore Management University, Singapore, Singapore