List of Abbreviations aint A = Ai — algebraic interior (or core) of the set A A∞ — asymptotic cone of the nonempty set A A⊥B — orthogonal closed linear subspaces. H = A⊕B — complementary subspaces, see under line (4.24) a.c. — asymptotically compact A∗ — adjoint operator to A AT — adjoint operator in Hilbert spaces (AT )−1 — inverse operator of the adjoint operator AT to A lin A — linear hull of a nonempty set A aff A — affine hull of a nonempty set A conv A — convex hull of a nonempty set A A + B — addition of sets, see Section 2.1 [A]C — see full set, Definition 2.1.21 Ac — Y \ A, where A ⊂ Y Ar — see Theorem 3.10.12 Ax — {y ∈ Y | (x, y) ∈ A}, where A ⊂ X × Y is nonempty a.s.P . — almost surely w.r.t. probability P α(y) — sup-support function B(x) — neighborhood base of x B0 — unit ball B(x, r) — (closed) ball of center x and radius r BEff(D; C) — the set of Benson-efficient points of D B-function — Bregman function cl B or B — closure of the set B τ-clB — closure of the set B w.r.t. the topology τ ∆ ◦ Γ — composition of multifunctions ∆ and Γ ∂B — B \ int B, see Section 3.3 bd B — usual (topological) boundary of B β(y) — inf-support function Cε — Henig dilating cone C-l.c. — C-lower continuous C-l.s.c. — C-lower semicontinuous C+ — continuous dual cone of the cone C infC A — infimum of A w.r.t. C C# — quasi interior of the cone C+ C[0, 1] — see Example 2.1.12 C1[0,T] — see inverse Stefan problem ≤C — see Section 3.2.3 k0 — a preorder, see Section 3.10 and Theorem 3.10.7 C-seq-b-regular — see under Corollary 3.10.16 (CP) — containment property C : X ⇒ Y — multifunction, whose values are pointed convex cones Cc — see Section 3.5.2 320 List of abbreviations CP — complementarity problem |P | — cardinality of P D⊥ — orthogonal space to a linear subspace D ≤ ∂ f(x0) — subdifferential of f at x0 ∈ dom f ≤ ∂ C f(x0) — subdifferential of f at x0 ∈ dom f dim — dimension DΓ(x, y)(u) — Dini upper derivative of Γ at (x, y) in the direction u DΓ (x, y)(u) — Dini lower derivative of Γ at (x, y) in the direction u SΓ(x, y)(u) — derivative in the sense of Shi in the direction u f (x)(v) — directional derivative of f at x in the direction v dist(x, A), d(x, A) — infa∈A d(x, a), where A ⊂ X and d a metric on X dom f — domain of f (DP) — domination property ϕcl B,k0 (y) — scalarization functional, see Section 2.3 Eff(M,Bεk0 ) — the set of approximately efficient points ε-Effϕ(M,C) — see Definition 3.1.2 EffMin(P, C) — see Remark 2.1.3 or Section 3.7 EffMax(D, C) — see Remark 2.1.3 or Section 3.7 EffMin(F, B, e)—(B,e)-minimal elements; see Section 3.11.2 EffMax(F, B, e) — the set of all (B,e)-maximal elements of F ∅ — empty set EP — equilibrium problem EVP — Ekeland’s variational principle epi f — epigraph of f e(A, B) — excess of A over B yε — see Section 3.1.1 E — expectation operator Ft — σ-algebra F(M) — the family of all nonempty finite subsets of M F(M,x) — the family of all finite subsets of M containing x ∇xf(x)—Fr´echet derivative of f w.r.t. x gr Γ — graph (of a multifunction Γ ) ΓC — see Section 2.4 Γf,C — see Section 2.4 Γ P — see Theorem 3.5.12 GEff(F) — the set of properly efficient points in the sense of Geoffrion GVEP — generalized vector equilibrium problem GVVI — generalized vector variational inequality HEff(D; C) — the set of Henig-proper efficient points of D w.r.t. C H-C-u.c. — Hausdorff C-upper continuous (w.r.t. cone C) H.l.c.s. — Hausdorff locally convex space H-l.c. — Hausdorff lower continuous H.t.v.s. — Hausdorff topological vector space H-u.c. — Hausdorff upper continuous HVIS — hemivariational inequalities system; see line (3.101) List of abbreviations 321 Im Γ — image (of a multifunction Γ ) χD — indicator function of D int C — interior of the set C Γ +1(B) — inverse image; see Section 2.4 Γ −1(B) — inverse image; see Section 2.4 ∞ — element adjoined to Y to get Y • Ker A — kernel of an operator A KKM-lemma — Knaster, Kuratowski, and Mazurkiewicz lemma (K, L)-monotonicity — see Definition 3.8.5 Φ(x, z∗) — Lagrangian (see Remark 3.11.5) Λα for α ∈ ]0, 1[ — see under Definition 2.4.1 ∞, p — see Example 2.2.3 p L or Lp — see Example 2.2.3 lim infx→x0 Γ (x) — limit inferior of Γ at x0 L(X, Y ), L(X, Y ) — the set of linear continuous mappings from X in Y L1(xε), L2(xε) — see Section 4.1.2 levΓ (y) — sublevel set; see Section 2.4 < levΓ (y) — strict sublevel set; see Section 2.4 l.c. — lower continuous l.c.s. — locally convex space l.s.c. — lower semicontinuous (X, P) — l.c.s. with a family P of seminorms + 2 — the common ordering cone in the Hilbert space 2 meas A — measure of a set A Γ : X ⇒ Y — multifunction Max(M0, R) — the class of maximal elements of M0 with respect to R Min(M0, R) — the class of minimal elements of M0 with respect to R HMax(Y ; C) — see Section 3.2.6 PrMax(Y ; C) — see Section 3.2.6 (xi)i∈I — net or generalized sequence Nτ (x) — the class of all neighborhoods of x w.r.t. a topology τ N — the set of nonnegative integers N∗ — the set of positive integers · ∗ — dual norm to · ||| · ||| — vector-valued norm 0 0 ND(x ) — normal cone to D at x ∈ X (X, · ) — normed space X n.v.s. — normed vector space (a, b) — open interval [a, b) — half open interval P(X) — the class of subsets of X p-Eff(F, Z) — the set of properly efficient points w.r.t. a family Z P -pseudomonotonicity — see Lemma 3.8.29 PrEff(Y ; C) — the set of properly efficient points of Y w.r.t. C PC r (r ≥ 1) — see Section 4.4.3 322 List of abbreviations pA — Minkowski functional of the set A U 0 — polar of the set U PX — projection of X × Y onto X PD(e) — projection of e onto a set D Pu — probability measure; see Section 4.5.3 R — R ∪{+∞} ∪ {−∞} R• — R ∪{+∞} R(PA) — range of the operator PA raint — relative algebraic interior (of a set) r : X × X → P —aP -valued metric, P a cone Rn — the real n-dimensional vector space R+ := [0, ∞[ — the set of nonnegative real numbers n n R+ — usual ordering cone of R R−1 — inverse of the relation (or multifunction) R + ⊂ ∈ Yt (x) — upper section of Y X with respect to t and x X − ⊂ ∈ Yt (x) — lower section of Y X with respect to t and x X S◦R — composition of two relations R and S — compare “directed set” X FSD Y — see Section 4.6 X SSD Y — see Section 4.6 SEP — scalar equilibrium problem SGVEP — strong generalized vector equilibrium problem S(f,P) — the set of solutions of a GVEP (cn) → c — sequence (cn) converging to c M × M — set of ordered pairs of elements of M (M,R) — set M with order structure R Min(F, y∗,e) — the set of all (y∗,e)-minimal elements of F Max(F, y∗,e) — the set of all (y∗,e)-maximal elements of F s.t. — subject to SPEff(A; C) — class of strong proper efficient points of set A w.r.t. C SVCP — strong vector complementarity problem TA — partial inverse of T with respect to A TB(A; a) — Bouligand tangent cone (or contingent cone) of A at a TU (A; a) — Ursescu tangent cone (or adjacent cone) of A at a t.l.s. (or t.v.s.) — topological linear (or topological vector) space τ (xi) → x — convergence (w.r.t. topology τ) τ1 × τ2 — product topology aT — transposed vector to a ∈ Rn • U — U \{x0}, where U is a neighborhood of x0 u.c. — upper continuous VOP — vector optimization problem VSP — vector saddle point problem VVI — vector variational inequality w-pointedness — see Section 3.8.1 w := σ(X, X∗) — weak topology of X List of abbreviations 323 w∗ := σ(X∗,X) — weak∗ topology of X∗ w-normal — weakly normal w xi x — weakly convergent net w.r.t. — with respect to WVCP — weak vector complementarity problem WVEP — weak vector equilibrium problem WVSP — weak vector saddle point problem WGVEP — weak generalized vector equilibrium problem. w Eff(A; C) — set of weakly efficient points X∗ — continuous dual space to the space X (X, ρ) — metric space with metric ρ (X, τ) — topological space X equipped with topology τ τ-a.c. set — asymptotically compact set w.r.t. topology τ x∗(x)=x, x∗ — value of linear continuous functional x∗ at x (x | y) — inner product in a Hilbert space Y • — Y ∪ {∞}, see Section 2.4 (Y,CY ) — linear space Y ordered by a cone CY References 1. Amahroq, T., Taa, A.: On Lagrange–Kuhn–Tucker multipliers for multiobjec- tive optimization problems.