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Merit functions: a bridge between optimization and equilibria

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Abstract

In the last decades, many problems involving equilibria, arising from engineering, physics and economics, have been formulated as variational mathematical models. In turn, these models can be reformulated as optimization problems through merit functions. This paper aims at reviewing the literature about merit functions for variational inequalities, quasi-variational inequalities and abstract equilibrium problems. Smoothness and convexity properties of merit functions and solution methods based on them will be presented.

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Notes

  1. The convergence is said global if it does not depend on the choice of the starting point.

  2. A sequence \(\{x^k\}\) is said to be convergent to \(\bar{x}\) with rate of convergence equal to \(r\) if

    $$\begin{aligned} \limsup _{k\rightarrow +\infty }\frac{\Vert x^{k+1} -\bar{x}\Vert }{\Vert x^{k} -\bar{x}\Vert ^r}=\gamma \in (0,+\infty ). \end{aligned}$$

    If \(r=1\) and \(\gamma \in (0,1)\), then the convergence is said to be linear, if \(r>1\), then the convergence is said to be superlinear, and, in particular, if \(r=2\), the convergence is said to be quadratic.

  3. The Armijo inexact line search along the direction \(d^k\) consists in finding the smallest non negative integer \(m\) such that

    $$\begin{aligned} p_{\alpha }(x^k+\beta ^m d^k) \le p_{\alpha }(x^k) - \sigma \,\beta ^m \Vert d^k\Vert ^2, \end{aligned}$$

    where \(\beta ,\sigma \in (0,1)\) are parameters, and then setting \(x^{k+1}:=x^k+\beta ^m d^k\).

  4. \(\delta _C\) is defined as follows: \(\delta _C(x)=0\) if \(x \in C\) and \(\delta _C(x)=+\infty \) otherwise.

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  1. Dipartimento di Informatica, Università di Pisa, Largo B. Pontecorvo 3, 56127 , Pisa, Italy

    Massimo Pappalardo, Giandomenico Mastroeni & Mauro Passacantando

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  1. Massimo Pappalardo
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  2. Giandomenico Mastroeni
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  3. Mauro Passacantando
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Pappalardo, M., Mastroeni, G. & Passacantando, M. Merit functions: a bridge between optimization and equilibria. 4OR-Q J Oper Res 12, 1–33 (2014). https://doi.org/10.1007/s10288-014-0256-5

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