(169b) A Software Framework for the Global Optimization of Nonconvex Two-Stage Stochastic Programs

Kannan, R., Massachusetts Institute of Technology
Barton, P. I., Massachusetts Institute of Technology

Most real-life optimization models
inherently contain uncertain model parameters. Stochastic programming provides
a natural way of addressing such formulations, and has been extensively used in
the process systems engineering literature [1-5].

This work presents a software framework
for solving two-stage stochastic programs with recourse, whose deterministic
equivalent form can be written as:

where , , , and functions , , and  are continuous.
The uncertainties are modeled using a finite number, s, of scenarios
indexed by h, each with a positive probability of occurrence . The binary and
continuous first-stage decisions y and z, respectively, are made
before the realization of the uncertainties, while the mixed-integer
second-stage decisions, , are made after
the realization of scenario h. Joint polyhedral constraints in y
and z are assumed (purely for notational convenience) to be modeled by
the scenario constraints in Problem (P).

Duality-based decomposition techniques
such as Benders decomposition [6], and its nonlinear variant generalized
Benders decomposition [7] can be used to solve Problem (P) in a decomposable
manner under certain restrictions on the participating functions. These
methods, however, rely on strong duality for convergence, which is not
guaranteed for most nonconvex problems. When the first-stage variables in
Problem (P) correspond purely to 0-1 decisions, nonconvex generalized Benders
decomposition (NGBD) can be used [2-4, 8]. Nonconvex generalized Benders
decomposition is an extension of generalized Benders decomposition that
rigorously handles nonconvexities in the participating functions in Problem (P).
A popular decomposition technique that can handle the general form of Problem
(P) is Lagrangian relaxation [9, 10]. The lower bounding problem in the
conventional Lagrangian relaxation technique involves the (partial) solution of
the Lagrangian dual problem, which is usually computationally intensive. Our
recent work [11] proposed a modified Lagrangian relaxation technique which
combines Lagrangian relaxation with NGBD and bounds tightening techniques to
solve Problem (P) when continuous first-stage decisions are to be made.

In this work, we provide a computational
framework for solving Problem (P) which implements versions of all of the
aforementioned decomposition techniques. We demonstrate the capabilities of our
framework via several process systems engineering-related case studies.

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algorithm for general nonconvex two-stage stochastic programs. In