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Nonlinear chance-constrained problems with applications to hydro scheduling

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Abstract

We present a Branch-and-Cut algorithm for a class of nonlinear chance-constrained mathematical optimization problems with a finite number of scenarios. Unsatisfied scenarios can enter a recovery mode. This class corresponds to problems that can be reformulated as deterministic convex mixed-integer nonlinear programming problems with indicator variables and continuous scenario variables, but the size of the reformulation is large and quickly becomes impractical as the number of scenarios grows. The Branch-and-Cut algorithm is based on an implicit Benders decomposition scheme, where we generate cutting planes as outer approximation cuts from the projection of the feasible region on suitable subspaces. The size of the master problem in our scheme is much smaller than the deterministic reformulation of the chance-constrained problem. We apply the Branch-and-Cut algorithm to the mid-term hydro scheduling problem, for which we propose a chance-constrained formulation. A computational study using data from ten hydroplants in Greece shows that the proposed methodology solves instances faster than applying a general-purpose solver for convex mixed-integer nonlinear programming problems to the deterministic reformulation, and scales much better with the number of scenarios.

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Acknowledgements

The authors are extremely grateful to Costas Baslis and Anastasios Bakirtzis for sharing the data on the Greek power system discussed in [22], to Alberto Borghetti for several helpful discussions, to the anonymous referees and the associate editor for their comments that helped significantly improve the paper. The first and fourth authors acknowledge the support of MIUR, Italy, under the Grant PRIN 2012. The second author acknowledges the support of the Air Force Office of Scientific Research under Award Number FA9550- 17-1-0025. Traveling support by the EU ITN 316647 “Mixed-Integer Nonlinear Optimization” is acknowledged by the third author. Part of this research was carried out at the Singapore University of Technology and Design, supported by grant SRES11012 and IDC grant IDSF1200108.

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A Analysis of the nonconvex formulation for scenario subproblems

A Analysis of the nonconvex formulation for scenario subproblems

In this “Appendix” we analyze the set \(C_x(w^i)\) when using the nonconvex generation function (23) for the scenario subproblems. More specifically, we show that this set satisfies the convexity assumption of Theorem 2.

Recall that in the language of this paper, \(C_x(w^i) = \text {Proj}_x C_{x,y}(w^i)\), and \(C_{x,y}(w^i)\) is defined by the constraints (8)–(14), with decision variables labeled \(x_{th}, w_{th}, e_{th}\) in the formulation. As discussed in Sect. 2, the master problem is only aware of variables \(x_{th}\). We therefore have to show that the projection of (8)–(14) onto the space of \(x_{th}\) is a convex set [while the variables labeled \(w_{th}, e_{th}\) are the y component of \(C_{x,y}(w^i)\)]. It is sufficient to show that this is true for a single constraint of the form (12), because all the remaining constraints are linear, and each of the Tn constraints (12) involves a different set of variables \(x_{th}, w_{th}, e_{th}\).

Using the generation function (23), constraint (12) can be written as follows:

$$\begin{aligned} f(x,w,e) = e + w(ax^2 - bx) + cx^2 -dx \le 0, \end{aligned}$$
(32)

where all coefficients abcd are positive in our data. Here, we dropped the subscripts from \(x_{th}, w_{th}, e_{th}\) for ease of exposition. Define \(F := \{(x,w,e) : f(x,w,e) \le 0\}\). Since the function \(e + w(ax^2 - bx) + cx^2 -dx\) is not convex in its arguments xwe, we show convexity of \(\text {Proj}_x F\) directly using the definition. Consider two points \(x_1, x_2 \in \text {Proj}_x F\), and we want to show \(x_3 := \lambda x_1 + (1-\lambda ) x_2 \in \text {Proj}_x F\) for any \(\lambda \in [0,1]\). We will show it under the additional conditions \(0 \le x \le \frac{b}{a}, w \ge 0\), both of which are satisfied by the data used in our experiments (we only consider nonnegative water levels, and the coefficients ab in the problem data are such that \(\frac{b}{a}\) is larger than the upper bound on x).

By convexity of the expression \(ax^2 - bx\) for \(a \ge 0\), we have \(ax_3^2 - bx_3 \le \lambda (ax_1^2 - bx_1) + (1-\lambda ) (ax_2^2 - bx_2)\) and \(cx_3^2 - dx_3 \le \lambda (cx_1^2 - dx_1) + (1-\lambda )( cx_2^2 - dx_2)\) for any \(\lambda \in [0,1]\). Choose \(w_3 = \max \{w_1, w_2\} \ge 0\), \(e_3 = \min \{e_1, e_2\}\). Then we have:

$$\begin{aligned} f(x_3, w_3, e_3)&= e_3 + w_3(ax_3^2 - bx_3) + cx_3^2 -dx_3 \\&\le e_3 + w_3 [\lambda (ax_1^2 - bx_1) + (1-\lambda ) (ax_2^2 - bx_2)] \\&\quad +\lambda (cx_1^2 - dx_1) + (1-\lambda )( cx_2^2 -dx_2) \\&= e_3 + \lambda [w_3 (ax_1^2 - bx_1) + cx_1^2 - dx_1] \\&\quad + (1-\lambda ) [w_3 (ax_2^2 - bx_2) + cx_2^2 - dx_2] \\&\le e_3 + \lambda [w_1 (ax_1^2 - bx_1) + cx_1^2 - dx_1] \\&\quad +(1-\lambda ) [w_2 (ax_2^2 - bx_2) + cx_2^2 - dx_2], \end{aligned}$$

where for the last inequality we used \(w_3 (ax_1^2 - bx_1) \le w_1 (ax_1^2 - bx_1)\) (because \(w_3 \ge w_1\) and \(ax_1^2 - bx_1 \le 0\), since \(0\le x \le \frac{b}{a}\)), and for similar reasons, \(w_3 (ax_2^2 - bx_2) \le w_2 (ax_2^2 - bx_2)\). Now consider the case \(w_1 (ax_1^2 - bx_1) + cx_1^2 - dx_1 \ge w_2 (ax_2^2 - bx_1) + cx_2^2 - dx_2\) first. Then we have:

$$\begin{aligned} f(x_3, w_3, e_3)&\le e_3 + \lambda [w_1 (ax_1^2 - bx_1) + cx_1^2 - dx_1] \\&\quad + (1-\lambda ) [w_2 (ax_2^2 - bx_1) + cx_2^2 - dx_2] \\&\le e_3 + w_1 (ax_1^2 - bx_1) + cx_1^2 - dx_1 \\&\le e_1 + w_1 (ax_1^2 - bx_1) + cx_1^2 - dx_1 \le 0, \end{aligned}$$

where for the second inequality we used \((1-\lambda ) [w_2 (ax_2^2 - bx_1) + cx_2^2 - dx_2] \le (1-\lambda )[w_1 (ax_1^2 - bx_1) + cx_1^2 - dx_1]\) since \(1-\lambda \ge 0\), and for the last inequality we used the fact that \(e_3 = \min \{e_1, e_2\} \le e_1\).

The case \(w_1 (ax_1^2 - bx_1) + cx_1^2 - dx_1 < w_2 (ax_2^2 - bx_1) + cx_2^2 - dx_2\) is almost identical and yields

$$\begin{aligned} f(x_3, w_3, e_3)&\le \min \{e_1, e_2\} + w_2 (ax_2^2 - bx_2) + cx_2^2 - dx_2 \\&\le e_2 + w_2 (ax_2^2 - bx_2) + cx_2^2 - dx_2 \le 0. \end{aligned}$$

This shows that \(x_3 \in \text {Proj}_x F\) for any \(\lambda \in [0,1]\), thereby proving convexity of \(\text {Proj}_x F\) and of the projection of the feasible set (8)–(14) with generation function (23). As a consequence, the first assumption of Theorem 2 is satisfied.

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Lodi, A., Malaguti, E., Nannicini, G. et al. Nonlinear chance-constrained problems with applications to hydro scheduling. Math. Program. 191, 405–444 (2022). https://doi.org/10.1007/s10107-019-01447-3

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