In this paper, we propose a new primal-dual algorithm for minimizing f(x)+g(x)+h(Ax), where f, g, and h are convex functions, f is differentiable with a Lipschitz continuous gradient, and A is a bounded linear operator. It has some famous primal-dual algorithms for minimizing the sum of two functions as special cases. For example, it reduces … Read more
We present an optimization method for Lipschitz continuous, piecewise smooth (PS) objective functions based on successive piecewise linearization. Since, in many realistic cases, nondifferentiabilities are caused by the occurrence of abs(), max(), and min(), we concentrate on these nonsmooth elemental functions. The method’s idea is to locate an optimum of a PS objective function by … Read more
We describe the fundamentals of algorithms for minimizing a smooth nonlinear function, and extensions of these methods to the sum of a smooth function and a convex nonsmooth function. Such objective functions are ubiquitous in data analysis applications, as we illustrate using several examples. We discuss methods that make use of gradient (first-order) information about … Read more
In this paper, we analyze the convergence of the alternating direction method of multipliers (ADMM) for minimizing a nonconvex and possibly nonsmooth objective function, $\phi(x_0,\ldots,x_p,y)$, subject to coupled linear equality constraints. Our ADMM updates each of the primal variables $x_0,\ldots,x_p,y$, followed by updating the dual variable. We separate the variable $y$ from $x_i$’s as it … Read more
The Mesh Adaptive Direct Search algorithm (MADS) is an iterative method for constrained blackbox optimization problems. One of the optional MADS features is a versatile search step in which quadratic models are built leading to a series of quadratically constrained quadratic subproblems. This work explores different algorithms that exploit the structure of the quadratic models: … Read more
In this paper we analyze the rate of local convergence of the augmented Lagrange method for solving optimization problems with equality constraints and the objective function expressed as the sum of a convex function and a twice continuously differentiable function. The presence of the non-smoothness of the convex function in the objective requires extensive tools … Read more
Locally weighted regression combines the advantages of polynomial regression and kernel smoothing. We present three ideas for appropriate and effective use of LOcally WEighted Scatterplot Smoothing (LOWESS) models for surrogate optimization. First, a method is proposed to reduce the computational cost of LOWESS models. Second, a local scaling coefficient is introduced to adapt LOWESS models … Read more
Pointwise and ergodic iteration-complexity results for the proximal alternating direction method of multipliers (ADMM) for any stepsize in $(0,(1+\sqrt{5})/2)$ have been recently established in the literature. In addition to giving alternative proofs of these results, this paper also extends the ergodic iteration-complexity result to include the case in which the stepsize is equal to $(1+\sqrt{5})/2$. … Read more
The extended second order cones were introduced by S. Z. Németh and G. Zhang in [S. Z. Németh and G. Zhang. Extended Lorentz cones and variational inequalities on cylinders. J. Optim. Theory Appl., 168(3):756-768, 2016] for solving mixed complementarity problems and variational inequalities on cylinders. R. Sznajder in [R. Sznajder. The Lyapunov rank of extended … Read more
Crouzeix’s conjecture states that for all polynomials p and matrices A, the inequality ||p(A)||