Figure 15 illustrates the LS function and its levels for the optimal geometry, i.e. for the solution problem.
In this paper, we investigate the solution problem of equilibrium problem (2.1) based on a projection algorithm.
Analogously, one can prove the uniqueness of the solution problem T1 in the case where λ 1 > 0, λ 2 < 0. □.
By dropping the constant terms which do not affect the solution, problem (37) can be written as minimize ∑ n ∈ N u n iT ( x n i + 1 - z n ) + v n i ( α n i + 1 - γ ) + ρ 2 ∥ x n i + 1 - z n ∥ 2 2 + ρ 2 ( α n i + 1 - γ ) 2, (48).
This yields the estimate biglvert u_{1}(r,varphi,z bigrvert < max Bigl{ max _{overline {D}}biglvert f_{1}(r,varphi bigrvert, max_{overline{D}}biglvert f_{2}(r,varphi bigrvert Bigr},quad (r,varphi,z in Q_{0}, (35) which implies the uniqueness of <span class="lh">the solution Problem D1.
In order to study the solution problem of equilibrium problem (2.1), we assume that f satisfies the following conditions: (A1) f ( x, x ) = 0, ∀ x ∈ C ; (A2) f is monotone, i.e., f ( x, y ) + f ( y, x ) ≤ 0, ∀ x, y ∈ C ; (A3) lim sup t ↓ 0 f ( t z + ( 1 − t ) x, y ) ≤ f ( x, y ), ∀ x, y, z ∈ C ; (A4) for each x ∈ C, y ↦ f ( x, y ) is convex and weakly lower semi-continuous.
