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Interior Proximal Method Without the Cutting Plane Property

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Abstract

A new interior proximal method for variational inequalities with generalized monotone operators is developed. It transforms a given variational inequality (which, maybe, is constrained and ill-posed) into unconstrained and well-posed equations as well as, at each iteration, one single additional extragradient step with rather small numerical efforts. Convergence is established under mild assumptions: The frequently assumed maximal monotonicity is weakened to pseudo- and quasimonotonicity with respect to the solution set, and a wide class of even nonlinearly constrained feasible sets is allowed for. In this general setting, the presented scheme constitutes the first interior proximal method that works without the so-called cutting plane property. Such a demanding assumption is completely left out, which allows to solve, e.g., wide classes of saddle point and equilibrium problems by means of an interior proximal method for the first time. As another application, we study variational inequalities derived from quasiconvex optimization problems.

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Notes

  1. Notions of monotonicity are introduced below.

  2. The reason is that in the sequel, we will work with multiple subsequences, i.e., a subsequence of a subsequence of..., and to this end, a notation as \(x^{k_{l_{m_{\ldots }}}}\) is rather complicated.

  3. Even if \(T\) is maximal monotone, our analysis provides new results. For instance, the feasible set may significantly more general than \(K=\mathbb {R}^n_+\), as in [5].

  4. It might be of interest that (7) can also be equivalently written as a variational inequality (see [16, Lemma 3.5]): Find \(y^k \in \mathbb {R}^n\) such that, for some error parameter \(\delta _k \ge \Vert e^k\Vert \),

    $$\begin{aligned} \langle t^k(y^k) + \chi _k \left( \nabla h(y^k) - \nabla h(x^k)\right) , x-y^k\rangle \ge -\delta _k \Vert x-y^k\Vert , \quad \forall ~ x \in K. \end{aligned}$$
    (8)

  5. It is shown later that our assumptions indeed ensure that \(\{x^k\} \subset \mathrm {int}K\).

  6. Burachik and Dutta worked with \(\vartheta (x) = c_1 x\) for some \(c_1 > 0\). But it might also be interesting to work with, e.g., \(\vartheta (x) = \sqrt{x}\). However, apart from that slight difference, the error criterion is exactly the same as in [5].

  7. Necessary changes for quasimonotonicity are discussed once convergence is established for the pseudomonotone case.

  8. The case \(\rho = + \infty \) is not excluded; we comment on this soon. Anyway, the subsequent analysis works regardless of whether \(\rho < + \infty \) or not.

  9. If \(T\) is monotone, we identify \(L = 0\) and define \(L/\kappa :=0\), regardless of whether \(\kappa = 0\).

  10. Loosely speaking, in view of Theorem 4.1 below this indicates that this function \(\varphi \) should, in general, only be applied if \(K\) is polyhedral.

  11. We mention that, for \(K=\mathbb {R}^n_+\), we recover the assumption \(r_k \le \min _{i=1,\ldots ,n} x_i\) as worked with in [5].

  12. Following a combination of the references [9, 23, 28], other examples may also be deduced, e.g., for \(K\) being a ball. Of course, in case of more complex and / or unbounded feasible sets \(K\) it is rather unlikely that \(\nabla h\) is explicitly invertible. Anyway, the extragradient step then still appears to be numerically attractive due to the remaining properties of \(\nabla h\), e.g., strict monotonicity and zone coerciveness.

  13. Note that in the proof of Theorem 5.2 no special property of \(T\) (in particular, no cutting plane property nor any related assumption) is required.

  14. In combination with Lemma 4.1, this establishes what has been announced above: If \(\overline{x} \in \mathrm {int}K\), and regardless of whether \(e^k \rightarrow 0\), then by Lemma 4.1, \(g_j(\overline{x}) < 0\) for all \(j=1,\ldots ,m\). Hence, \(I(\overline{x}) = \emptyset \) and the sum of the inferior limits is zero. In that case, no further argumentation is necessary. It is therefore enough to consider \(i \in I(\overline{x})\) only.

  15. The remaining \(i \in I(\overline{x})\) will be dealt with subsequently in an iterative manner.

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  1. Universität Trier, 54286 , Trier, Germany

    Nils Langenberg

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Langenberg, N. Interior Proximal Method Without the Cutting Plane Property. J Optim Theory Appl 166, 529–557 (2015). https://doi.org/10.1007/s10957-014-0605-8

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