Without requiring the strict complementarity, it is proved that, under second order sufficiency conditions, the augmented Lagrangian function admits a local saddle point. The existence of global saddle points is then obtained under additional assumptions that do not require the compactness of the ...
This paper describes a new method for constrained optimization which obtains its search directions from a quadratic programming subproblem based on the well-known augmented Lagrangian function. It avoids the penalty parameter to tend to infinity. We employ the Fletcher's exact penalty function as a ...
The augmented Lagrangian function is constructed as follows. 𝐿(𝐈,𝐗,𝐘,𝜇)=‖𝐈‖∗+𝜆‖𝐗‖1−〈𝐘,𝐈+𝐗−𝐒〉+𝜇2‖𝐈+𝐗−𝐒‖𝐹L(I,X,Y,μ)=‖I‖∗+λ‖X‖1−〈Y,I+X−S〉+μ2‖I+X−S‖F (13) where 𝜇μ is the ...
Thus, we have developed all equalities needed to compute the stochastic gradient of the augmented Lagrangian function (Eq. (15)) at each steepest ascent iteration (Eq. (4)) in the inner loop of the augmented Lagrange method. In all examples, we use a 1:1 ratio of control perturbations ...
which implies that the value of the augmented Lagrangian function will always decrease with the iteration progressing. We note that as long as parameter γ2≠0, one can always find a suitable ρ large enough such that the condition ργ2>2K2 is satisfied, since ργ2 is monotonically increas...
First, the alternating direction method of multipliers (ADMM) is extended, assuming that it is easy to optimize the augmented Lagrangian function with one block of variables at each time while fixing the other block. We prove that iteration complexity bound holds under suitable conditions, where t...
The equivalence between an η-saddle point of the η-Lagrangian of the associated η-approximated optimization problem and an optimal solution in the ... T.,Antczak - 《Journal of Optimization Theory & Applications》 被引量: 12发表: 2007年 A saddle point of Lagrange function and unsmooth penal...
In this paper we consider an augmented Lagrangian method for the minimization of a nonlinear functional in the presence of an equality constraint whose image space is in a Hilbert space, an inequality constraint whose image space is finite dimensional, and an affine inequality constraint whose image...
t.p=swhereg(s)=0ifs⩾0+∞ifs<0Then, the augmented Lagrangian functionLρ(p,s,w)=12p′Hp-u¯′p+g(s)+w′(p-s)+ρ2‖p-s‖22is considered, where ρ>0 is a parameter of the algorithm. The basic ADMM algorithm consists of the following iterations:(22)pk+1=argminpLρ(p,...
Finally, we point out that the augmented Lagrangian function \(f(x)+\lambda^{T}(x-y)+\frac{\rho}{2}\|x-y\|^{2}_{2}\) can be also employed in the penalized problem (28). 6 Conclusions We have summarized in this paper some recent advances in mathematical programming with semi-...