Maximally monotone linear subspace extensions of monotone subspaces: explicit constructions and characterizations

Abstract

Monotone linear relations play important roles in variational inequality problems and quadratic optimizations. In this paper, we give explicit maximally monotone linear subspace extensions of a monotone linear relation in finite dimensional spaces. Examples are provided to illustrate our extensions. Our results generalize a recent result by Crouzeix and Anaya.2000 Mathematics Subject Classification: Primary 47H05; Secondary 47B65, 47A06, 49N15

On the construction of maximal $p$-cyclically monotone operators

“…Construction of maximal monotone operators was addressed previously by Crouzeix, Ocaña-Anaya and Sosa [6]. See also [11]. The goal of this work is to provide a way to construct explicit examples of maximal p-cyclically monotone operators.…”

“…• z 0 Psy 2 , y 3 r ðñ z 0 " p´t, 1 ´2t, ´tq, t Ps0, 1r. Replacing this in equations (11) Therefore w ´u " 16 so 16 `v ´4t ď w, u `v `2w ě 20 and T 5 pt, 0, 1 ´tq " tpu, v, wq : w ´u " 16, w ´v ě 16 ´4t, u `v `2w ě 20u " p´3 ´t, 3t ´3, 13 ´tq `cone cotp1, ´3, 1q, p1, 1, 1qu.…”

For maximally monotone linear relations, dense type, negative-infimum type, and Fitzpatrick-Phelps type all coincide with monotonicity of the adjoint

“…Moreover, we generalize the results to the linear relations. Linear relations have recently become a center of attention in Monotone Operator Theory; see, e.g., [1,2,4,5,6,7,8,9,16,17,27,34,35,36,37,38,39,40,41] and Cross' book [19] for general background on linear relations.…”