Abstract
We propose to solve a general quasi-variational inequality by using its Karush–Kuhn–Tucker conditions. To this end we use a globally convergent algorithm based on a potential reduction approach. We establish global convergence results for many interesting instances of quasi-variational inequalities, vastly broadening the class of problems that can be solved with theoretical guarantees. Our numerical testings are very promising and show the practical viability of the approach.
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References
Baillon, J.-B., Haddad, G.: Quelques propriétés des opérateurs angle-bornés et n-cycliquement monotones. Isr. J. Math. 26, 137–150 (1977)
Baiocchi, C., Capelo, A.: Variational and Quasivariational Inequalities: Applications to Free Boundary Problems. Wiley, New York (1984)
Bensoussan, A.: Points de Nash dans le cas de fonctionnelles quadratiques et jeux differentiels lineaires a N personnes. SIAM J. Control 12, 460–499 (1974)
Bensoussan, A., Goursat, M., Lions, J.-L.: Contrôle impulsionnel et inéquations quasi-variationnelles stationnaires. C. R. Acad. Sci. Paris Sér. A 276, 1279–1284 (1973)
Bensoussan, A., Lions, J.-L.: Nouvelle formulation de problèmes de contrôle impulsionnel et applications. C. R. Acad. Sci. Paris Sér. A 276, 1189–1192 (1973)
Bensoussan, A., Lions, J.-L.: Nouvelles méthodes en contrôle impulsionnel. Appl. Math. Optim. 1, 289–312 (1975)
Beremlijski, P., Haslinger, J., Kočvara, M., Outrata, J.: Shape optimization in contact problems with Coulomb friction. SIAM J. Optim. 13, 561–587 (2002)
Bliemer, M., Bovy, P.: Quasi-variational inequality formulation of the multiclass dynamic traffic assignment problem. Transp. Res. B Methodol. 37, 501–519 (2003)
Chan, D., Pang, J.-S.: The generalized quasi-variational inequality problem. Math. Oper. Res. 7, 211–222 (1982)
Cottle, R.W., Pang, J.-S., Stone, R.E.: The Linear Complementarity Problem. Academic Press, New York (1992)
De Luca, M., Maugeri, A.: Discontinuous quasi-variational inequalities and applications to equilibrium problems. In: Giannessi, F. (ed.) Nonsmooth Optimization: Methods and Applications, pp. 70–75. Gordon and Breach, Amsterdam (1992)
Dirkse, S.P., Ferris, M.C.: The PATH solver: a non-monotone stabilization scheme for mixed complementarity problems. Optim. Methods Softw. 5, 123–156 (1995)
Donato, M.B., Milasi, M., Vitanza, C.: An existence result of a quasi-variational inequality associated to an equilibrium problem. J. Glob. Optim. 40, 87–97 (2008)
Dreves, A., Facchinei, F., Kanzow, C., Sagratella, S.: On the solution of the KKT conditions of generalized Nash equilibrium problems. SIAM J. Optim. 21, 1082–1108 (2011)
Facchinei, F., Kanzow, C.: Penalty methods for the solution of generalized Nash equilibrium problems. SIAM J. Optim. 20, 2228–2253 (2010)
Facchinei, F., Kanzow, C.: Generalized Nash equilibrium problems. Ann. Oper. Res. 175, 177–211 (2010)
Facchinei, F., Kanzow, C., Sagratella, S.: QVILIB: A library of quasi-variational inequality test problems. To appear in Pac. J. Optim.
Facchinei, F., Pang, J.-S.: Finite-Dimensional Variational Inequalities and Complementarity Problems, vols. I and II. Springer, New York (2003)
Ferris, M.C., Munson, T.S.: Interfaces to PATH 3.0: design, implementation and usage. Comput. Optim. Appl. 12, 207–227 (1999)
Fukushima, M.: A class of gap functions for quasi-variational inequality problems. J. Ind. Manag. Optim. 3, 165–171 (2007)
Glowinski, R.: Numerical methods for variational inequalities. Springer, Berlin (2008, reprint of the 1984 edition)
Glowinski, R., Lions, J.L., Trémolières, R.: Numerical Analysis of Variational Inequalities. North-Holland, Amsterdam (1981)
Hammerstein, P., Selten, R.: Game theory and evolutionary biology. In: Aumann, R.J., Hart, S. (eds.) Handbook of Game Theory with Economic Applications, vol. 2, pp. 929–993. North-Holland, Amsterdam (1994)
Harker, P.T.: Generalized Nash games and quasi-variational inequalities. Eur. J. Oper. Res. 54, 81–94 (1991)
Haslinger, J., Panagiotopoulos, P.D.: The reciprocal variational approach to the Signorini problem with friction. Approximation results. Proc. R. Soc. Edinb. Sect. A Math. 98, 365–383 (1984)
Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press, Cambridge (1990)
Ichiishi, T.: Game Theory for Economic Analysis. Academic Press, New York (1983)
Kharroubi, I., Ma, J., Pham, H., Zhang, J.: Backward SDEs with constrained jumps and quasi-variational inequalities. Ann. Probab. 38, 794–840 (2010)
Kravchuk, A.S., Neittaanmaki, P.J.: Variational and Quasi-variational Inequalities in Mechanics. Springer, Dordrecht (2007)
Kubota, K., Fukushima, M.: Gap function approach to the generalized Nash equilibrium problem. J. Optim. Theory Appl. 144, 511–531 (2010)
Kunze, M., Rodrigues, J.F.: An elliptic quasi-variational inequality with gradient constraints and some of its applications. Math. Methods Appl. Sci. 23, 897–908 (2000)
Monteiro, R.D.C., Pang, J.-S.: A potential reduction Newton method for constrained equations. SIAM J. Optim. 9, 729–754 (1999)
Mosco, U.: Implicit variational problems and quasi variational inequalities. In: Gossez, J., Lami Dozo, E., Mawhin, J., Waelbroeck, L. (eds.) Nonlinear Operators and the Calculus of Variations, Lecture Notes in Mathematics, vol. 543, pp. 83–156. Springer, Berlin (1976)
Nesterov, Y., Scrimali, L.: Solving strongly monotone variational and quasi-variational inequalities. CORE discussion paper 2006/107, Catholic University of Louvain, Center for Operations Research and Econometrics (2006)
Noor, M.A.: On general quasi-variational inequalities. J. King Saud Univ. 24, 81–88 (2012)
Noor, M.A.: An iterative scheme for a class of quasi variational inequalities. J. Math. Anal. Appl. 110, 463–468 (1985)
Ortega, J.M., Rheinboldt, J.C.: Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, New York (1970). (reprinted by SIAM, Philadelphia (2000))
Outrata, J., Kocvara, M.: On a class of quasi-variational inequalities. Optim. Methods Softw. 5, 275–295 (1995)
Outrata, J., Kocvara, M., Zowe, J.: Nonsmooth Approach to Optimization Problems with Equilibrium Constraints. Kluwer, Dordrecht (1998)
Outrata, J., Zowe, J.: A Newton method for a class of quasi-variational inequalities. Comput. Optim. Appl. 4, 5–21 (1995)
Pang, J.-S.: The implicit complementarity problem. In: Mangasarian, O.L., Meyer, R.R., Robinson, S.M. (eds.) Nonlinear Programming 4, pp. 487–518. Academic Press, New York (1981)
Pang, J.-S.: On the convergence of a basic iterative method for the implicit complementarity problem. J. Optim. Theory Appl. 37, 149–162 (1982)
Pang, J.-S.: Error bounds in mathematical programming. Math. Program. 79, 299–332 (1997)
Pang, J.-S., Fukushima, M.: Quasi-variational inequalities, generalized Nash equilibria, and multi-leader-follower games. Computational Management Science 2, 21–56 (2005) [Erratum: ibid 6, 373–375 (2009)]
Ryazantseva, I.P.: First-order methods for certain quasi-variational inequalities in Hilbert space. Comput. Math. Math. Phys 47, 183–190 (2007)
Scrimali, L.: Quasi-variational inequality formulation of the mixed equilibrium in mixed multiclass routing games. CORE discussion paper 2007/7, Catholic University of Louvain, Center for Operations Research and Econometrics (2007)
Siddiqi, A.H., Ansari, Q.H.: Strongly nonlinear quasivariational inequalities. J. Math. Anal. Appl. 149, 444–450 (1990)
Wei, J.Y., Smeers, Y.: Spatial oligopolistic electricity models with Cournot generators and regulated transmission prices. Oper. Res. 47, 102–112 (1999)
Acknowledgments
We are thankful to Jiří Outrata and Michal Kočvara who very kindly provided us with the Matlab codes used to generate the data for the test problems OutKZ31 and OutKZ41.
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A Appendix on monotonicity and Lipschitz properties
A Appendix on monotonicity and Lipschitz properties
In this appendix we recall some well-known definitions and discuss some related results. Although the latter are also mostly well-known, in some cases we could not find in the literature the exact versions we needed. Therefore, for completeness we also report the proofs of these results.
We begin by recalling the definitions of several classes of functions.
Definition 2
Let \( D \subseteq {\mathbb{R }}^n \) and \( F: D \rightarrow {\mathbb{R }}^n \) be a given function. Then
-
(a)
\(F\) is strongly monotone on \(D\) with constant \(\sigma \) if \( \sigma >0 \) and
$$\begin{aligned} (x - y)^{\scriptscriptstyle T}\big ( F(x) - F(y) \big ) \,\ge \, \sigma \Vert x - y\Vert ^2, \quad \forall x, y \in D; \end{aligned}$$The largest \(\sigma \) for which such a relation holds is termed the monotonicity modulus of \(F\) on \(D\):
$$\begin{aligned} \sigma (D,F) \, :=\, \inf _{x\ne y, x,y \in D}\frac{(x - y)^{\scriptscriptstyle T}\big ( F(x) - F(y) \big )}{\Vert x-y\Vert ^2}. \end{aligned}$$ -
(b)
\(F\) is co-coercive on \(D\) with constant \(\xi \) if \( \xi >0 \) and
$$\begin{aligned} (x - y)^{\scriptscriptstyle T}\big ( F(x) - F(y) \big ) \,\ge \, \xi \Vert F(x) - F(y)\Vert ^2, \quad \forall x, y \in D; \end{aligned}$$ -
(c)
\(F\) is Lipschitz continuous on \(D\) with constant \(L \ge 0\) if
$$\begin{aligned} \Vert F(x) - F(y)\Vert \, \le \, L \Vert x - y\Vert , \quad \forall x, y \in D. \end{aligned}$$The smallest \(L\) for which such relation holds is termed the Lipschitz modulus of \(F\) on \(D\):
$$\begin{aligned} L(D,F) \, :=\, \sup _{x\ne y, x,y \in D} \frac{\Vert F(x) - F(y)\Vert }{\Vert x-y\Vert }. \end{aligned}$$ -
(d)
\(F\) is a homeomorphism of \(D\) onto \(F(D)\) if \(F\) is one-to-one on \(D\) (that is \(F(x) \ne F(y)\) whenever \(x,y \in D, x \ne y\), or, in other words, \(F\) has a single-valued inverse \(F^{-1}\) defined on \(F(D)\)), and \(F\) and \(F^{-1}\) are continuous on \(D\) and \(F(D)\), respectively. \(\square \)
Characterizations of the Lipschitz and strong monotonicity moduli are given in the following result.
Proposition 3
Let \( D \subseteq {\mathbb{R }}^n \) be an open, convex subset of \({\mathbb{R }}^n\) and let \( F: D \rightarrow {\mathbb{R }}^n \) be a continuously differentiable function. Then the following statements hold:
-
(a)
\(F\) is Lipschitz continuous on \( D \) with constant \(L\) if and only if \( \Vert J\!F(x)\Vert \le L \) for all \( x \in D \); consequently
$$\begin{aligned} L(D,F) \, =\, \sup _{x\in D}\Vert J\!F(x)\Vert , \end{aligned}$$provided the \(\sup \) on the right hand side is finite.
-
(b)
\(F\) is strongly monotone on \( D\) with constant \(\sigma \) if and only if \( h^{\scriptscriptstyle T}J\!F(x) h \ge \sigma \Vert h\Vert ^2 \) for all \( x \in D , \, h \in {\mathbb{R }}^n \); consequently
$$\begin{aligned} \sigma (D,F) \, =\, \inf _{x\in D}\mu _m^s(J\!F(x)), \end{aligned}$$provided the \(\inf \) on the right hand side is positive.
Proof
-
(a)
From Theorem 3.2.3 in [37], if \( \Vert J\!F(x)\Vert \le L \) then \(L\) is a Lipschitz constant for \(F\) on \(D\). Conversely, assume that
$$\begin{aligned} \big \Vert F(x) - F(y) \big \Vert \le L \Vert x - y \Vert , \quad \forall x, y \in D \end{aligned}$$(35)holds. Applying the differential mean value theorem to each component function \( F_i \) of \( F \), it follows that, for any given \( x, y \in D \), we can find suitable points \( \xi ^{(i)} \in (x,y) \) such that
$$\begin{aligned} F_i(x) - F_i (y) = \nabla F_i (\xi ^{(i)})^T (x-y) \quad \forall i = 1, \ldots , n. \end{aligned}$$Setting
$$\begin{aligned} G(\xi ) := \left( \begin{array}{ccc}&\nabla F_1 (\xi ^{(1)})^T&\\&\vdots&\\&\nabla F_n (\xi ^{(n)})^T&\end{array} \right) \in {\mathbb{R }}^{n \times n}, \end{aligned}$$this can be rewritten in a compact way as
$$\begin{aligned} F(x) - F(y) = G(\xi ) (x-y). \end{aligned}$$(36)Now, let \( x \in D \) be fixed, and note that
$$\begin{aligned} G( \xi ) \rightarrow J\!F (x) \end{aligned}$$(37)for any sequence \( y \rightarrow x \) in view of the continuous differentiability of \( F \). We now consider a particular sequence \( y = x + td \) with a fixed (but arbitrary) vector \( d \in {\mathbb{R }}^n \setminus \{ 0 \} \) and a sequence \( t \downarrow 0 \). Then (35) and (36) together imply
$$\begin{aligned} \big \Vert G(\xi ) t d \big \Vert = \big \Vert F(x) - F(x + td) \big \Vert \le L \Vert t d \Vert . \end{aligned}$$Dividing by \( t \) and subsequently letting \( t \downarrow 0 \) (note that \( \xi \) still depends on \( t \)), we obtain
$$\begin{aligned} \big \Vert J\!F(x) d \big \Vert \le L \Vert d \Vert \end{aligned}$$in view of (37). Since \( d \) was taken arbitrarily, this implies \( \Vert J\!F(x) \Vert \le L \), and this inequality is true for any vector \( x \in D \).
-
(b)
See [37, Theorem 5.4.3]. \(\square \)
The following result gives a relation between the Lipschitz constants etc. of a given mapping \( F \) and its inverse \( F^{-1} \).
Proposition 4
Let a function \(F: D\rightarrow {\mathbb{R }}^n\) be given where \(D\) is an open subset of \({\mathbb{R }}^n\). Assume that two positive constants \(\ell \) and \(L\) exist such that
Then \(F\) is a homeomorphism from \(D\) to \(F(D)\) (which is an open set) and
in particular, \( F \) and \( F^{-1} \) are Lipschitz continuous on \( D \) and \( F(D) \), respectively.
Proof
The first inequality from (38) implies that \(F\) is one-to-one on \(D\) (therefore the inverse \( F^{-1} \) exists on \( F(D)\)), and that, setting \( a = F (x)\) and \(b = F (y)\), the second inequality in (39) holds. In particular, this implies that \( F^{-1} \) is Lipschitz continuous on \( F(D) \), hence continuous, so that \( F(D) = (F^{-1})^{-1} (D) \), being the pre-image of a continuous map of the open set \( D \), is also an open set. Finally, let \( a, b \in F(D)\) be arbitrarily given. Setting \( x = F^{-1} (a),y = F^{-1} (b) \), we obtain from the second inequality in (38) that
and this completes the proof. \(\square \)
The next result considers a strongly monotone and Lipschitz continuous mapping and provides suitable bounds for the moduli of Lipschitz continuity and strong monotonicity of the corresponding inverse function. We stress, however, that the constant of strong monotonicity of the inverse function provided by this result is really just an estimate and typically not exact. It seems difficult to find a stronger bound in the general context discussed here. In a more specialized situation, much better results can be obtained, see Proposition 6 below.
Proposition 5
Let \( D \subseteq {\mathbb{R }}^n \) be an open set and \( F: D \rightarrow {\mathbb{R }}^n \) be strongly monotone with modulus \(\sigma \) and Lipschitz continuous with modulus \(L\) on \(D\). Then \(F\) is co-coercive with constant \(\frac{\sigma }{L^2}\). Furthermore, it holds that the inverse \( F^{-1} \) exists on \(F(D)\), is Lipschitz with constant \(\frac{1}{\sigma }\) and strongly monotone with constant \(\frac{\sigma }{L^2}\).
Proof
We can write
and
by assumption. A combination of these two inequalities yields
Hence \( F \) is co-coercive with constant \( \frac{\sigma }{L^2} \).
By Proposition 4 we know that \(F\) is a homeomorphism from \(D\) to \(F(D)\) and \(F^{-1}\) is Lipschitz continuous with constant \(\frac{1}{\sigma }\). Finally writing \( a=F(x), b=F(y) \) in (40) gives
This completes the proof. \(\square \)
The following result gives an exact estimate of the Lipschitz and strong monotonicity moduli of the inverse of a function under the assumption that the mapping \( F \) itself is a gradient mapping, i.e. that \( F = \nabla f \) for a differentiable real-valued function \( f \).
Proposition 6
Let \( D \subseteq {\mathbb{R }}^n \) be an open convex set and \(F: D\rightarrow {\mathbb{R }}^n\) be a gradient mapping. Assume that \(F\) is strongly monotone with modulus \(\sigma \) and Lipschitz continuous with modulus \(L\) on \(D\). Then the inverse function \( F^{-1} \) exists on \(F(D)\), is Lipschitz with modulus \(\frac{1}{\sigma }\) and strongly monotone with modulus \(\frac{1}{L}\).
Proof
The result can easily be derived from the Baillon–Haddad Theorem, see [1], when \(D={\mathbb{R }}^n\). We give here a direct proof which is valid also when \(D \ne {\mathbb{R }}^n\). In view of Proposition 5, we only have to verify the statement that \( F^{-1} \) is strongly monotone with constant \(\frac{1}{L}\).
To this end, first consider a symmetric positive definite matrix \( A \in {\mathbb{R }}^{n \times n} \), let \( A^{1/2} \) be the corresponding (unique) symmetric positive definite square root of \( A \) so that \( A^{1/2} A^{1/2} = A \), and let \( A^{-1/2} \) be the inverse of \( A^{1/2} \). Then the symmetry of \( A^{1/2} \) together with the Cauchy–Schwarz inequality implies
Squaring both sides shows that
holds. Since \( F \) is strongly monotone, the Jacobian \( J\!F(x) \) is positive definite for all \( x \in D \); furthermore, since \( F \) is a gradient mapping, this Jacobian is also symmetric. Hence we can apply inequality (41) to the matrix \( A := J\!F(x) \) and obtain
where the second inequality uses the Cauchy–Schwarz inequality once again, and the third inequality takes into account Proposition 3. This implies
Since \( J\!F(x)^{-1} = J\!F^{-1}(y) \) for \( y = F(x) \) by the Inverse Function Theorem, this gives
By a well-known result, see [37, Theorem 5.4.3] this is equivalent to saying that \( F^{-1} \) is strongly monotone on \( F(D) \) with constant \( 1/L \). \(\square \)
The sharper result from Proposition 6 regarding the modulus of strong monotonicity does, in general, not hold for non-gradient mappings, see the corresponding discussion and (counter-) example at the end of Sect. 3.2.
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Facchinei, F., Kanzow, C. & Sagratella, S. Solving quasi-variational inequalities via their KKT conditions. Math. Program. 144, 369–412 (2014). https://doi.org/10.1007/s10107-013-0637-0
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DOI: https://doi.org/10.1007/s10107-013-0637-0