The projection method for solving problem (1.1) came originally from the Goldstein (see [3]) and Levitin-Polyak (see [4]) gradient projection method for the box-constrained minimization and was studied by many researchers such as Auslender (see [5]), Bakusinskii-Polyak (see [6]), Bruck (see [7]), Noor-Wang-Xiu (see [8]) and Xiu-Wang-Zhang (see [9]).
Local deterministic and global probabilistic minimization methods are studied.
To demonstrate the validity and effectiveness of the proposed method, three compliance minimization problems are studied and their optimized solutions present significant mechanical benefits of incorporating the nonlinearities, in terms of remarkable enhancement in not only the structural stiffness but also the critical buckling load.
Numerical examples of mean compliance minimization in 2D are studied in regard to efficiency, convergence and accuracy.
The total MSE minimization criterion has been studied in [24, 27, 28] and can be expressed in terms of the Lagrangian dual objective function: L ϵ k, E k, λ = ∑ k = 1 K ∗ ϵ k + λ ∑ k = 1 K ∗ E k − E T (16).
Methods for solving the equilibrium problem and the constrained convex minimization problem have extensively been studied respectively in a Hilbert space.
In the past, both parametric and functional minimizations of engine irreversibility have been studied extensively.
}t in[0,T], Delta u'(t_{j} ) = I_{j} (u(t_{j} )),quad j = 1,2, ldots,n, u(0) = u(T), end{cases} is studied by the minimization and the mountain pass theorem.
The pioneering work in this direction is the paper of Nieto and O'Regan [5], where the second-order impulsive problem { − u ″ + λ u = f ( t, u ), t ≠ t j, a.e. t ∈ [ 0, T ], u ( 0 ) = u ( T ), Δ u ′ ( t j ) = I j ( u ( t j ) ), j = 1, 2, …, n. (with Δ u ′ ( t j ) : = u ′ ( t j + ) − u ′ ( t j − ) ) is studied, using the minimization and the mountain-pass theorem.
Systematic cutting process design and optimization problems are studied for surface roughness minimization by stochastic algorithms.
