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A bundle method for nonsmooth DC programming with application to chance-constrained problems

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Abstract

This work considers nonsmooth and nonconvex optimization problems whose objective and constraint functions are defined by difference-of-convex (DC) functions. We consider an infeasible bundle method based on the so-called improvement functions to compute critical points for problems of this class. Our algorithm neither employs penalization techniques nor solves subproblems with linearized constraints. The approach, which encompasses bundle methods for nonlinearly-constrained convex programs, defines trial points as solutions of strongly convex quadratic programs. Different stationarity definitions are investigated, depending on the functions’ structures. The approach is assessed in a class of nonsmooth DC-constrained optimization problems modeling chance-constrained programs.

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Acknowledgements

The authors are grateful to two anonymous referees for their constructive suggestions that improved the original version of this article. The fifth author acknowledges the partial financial support of PGMO (Gaspard Monge Program for Optimization and operations research) of the Hadamard Mathematics Foundation, through the project “Models for planning energy investment under uncertainty”.

Author information

Authors and Affiliations

  1. EDF R&D, Boulevard Gaspard Monge, Paris, France

    W. van Ackooij, P. Javal & B. Swaminathan

  2. MINES ParisTech, CMA – Centre de Mathématiques Appliquées, PSL – Research University, Sophia-Antipolis, France

    S. Demassey, P. Javal & W. de Oliveira

  3. INESC-ID, Department of Electrical and Computer Engineering, Instituto Superior Técnico-IST, Universidade de Lisboa, Lisbon, Portugal

    H. Morais

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  1. W. van Ackooij
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  2. S. Demassey
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  3. P. Javal
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  4. H. Morais
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  5. W. de Oliveira
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  6. B. Swaminathan
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Correspondence to P. Javal.

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Auxiliary results

Auxiliary results

The following simple lemma helps compute the subdifferential of \(F_{\tau ^{}}{}(x)=\max \{f_1(x)+c_2(x)- \tau _f,\,f_2( x)+c_1(x)- \tau _c\}\).

Lemma 3

Let \({{\bar{x}}} \in X\) and \({{\bar{\tau }}}=({{\bar{\tau }}}_f,{{\bar{\tau }}}_c)\) as in (25) be given.

  1. i)

    If \(c({{\bar{x}}})>0\), then \(f_1({{\bar{x}}})+c_2({{\bar{x}}})-\bar{\tau }_f<f_2({{\bar{x}}})+c_1({{\bar{x}}})- {{\bar{\tau }}}_c\).

  2. ii)

    If \(c({{\bar{x}}})< 0\), then \(f_1({{\bar{x}}})+c_2({{\bar{x}}})-{{\bar{\tau }}}_f> f_2({{\bar{x}}})+c_1({{\bar{x}}})- {{\bar{\tau }}}_c\).

  3. iii)

    If \(c({{\bar{x}}})= 0\), then \(f_1({{\bar{x}}})+c_2({{\bar{x}}})-{{\bar{\tau }}}_f = f_2(\bar{x})+c_1({{\bar{x}}})- {{\bar{\tau }}}_c\).

Proof

The definition of \({\bar{\tau }}=\big (f({{\bar{x}}})+\rho \max \{c({{\bar{x}}}),0\}, \sigma \max \{c({{\bar{x}}}),0\}\big )\) with \(\rho \ge 0\) and \(\sigma \in [0,1)\) gives:

$$\begin{aligned}&f_1({{\bar{x}}})+c_2({{\bar{x}}})-{{\bar{\tau }}}_f-(f_2({{\bar{x}}})+c_1({{\bar{x}}})- {{\bar{\tau }}}_c) \\&\quad = (\sigma -\rho )\max \{c({{\bar{x}}}),0\} -c({{\bar{x}}}) \\&\quad = {\left\{ \begin{array}{ll} (\sigma -1-\rho )c({{\bar{x}}}) < 0 &{}\quad \text {if } c({{\bar{x}}})>0\\ -c({{\bar{x}}}) \ge 0 &{}\quad \text {if } c({{\bar{x}}})\le 0. \end{array}\right. } \end{aligned}$$

\(\square \)

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van Ackooij, W., Demassey, S., Javal, P. et al. A bundle method for nonsmooth DC programming with application to chance-constrained problems. Comput Optim Appl 78, 451–490 (2021). https://doi.org/10.1007/s10589-020-00241-8

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