## Semi-Infinite Generalized Disjunctive and Mixed Integer Convex Programs with(out) Uncertainty

In this paper, we introduce semi-infinite generalized disjunctive programs that are defined by logical propositions along with disjunctions of sets of logical equations and infinite number of algebraic inequalities. We denote these programs by SIGDPs. For SIGDPs with linear and convex inequalities, we present new reformulations: semi-infinite mixed-binary/disjunctive linear programs and semi-infinite mixed-binary/disjunctive convex programs, … Read more

## On an iteratively reweighted linesearch based algorithm for nonconvex composite optimization

In this paper we propose a new algorithm for solving a class of nonsmooth nonconvex problems, which is obtained by combining the iteratively reweighted scheme with a finite number of forward–backward iterations based on a linesearch procedure. The new method overcomes some limitations of linesearch forward–backward methods, since it can be applied also to minimize … Read more

## A Novel Stepsize for Gradient Descent Method

In this paper, we propose a novel stepsize for the classical gradient descent scheme to solve unconstrained nonlinear optimization problems. We are concerned with the convex and smooth objective without the globally Lipschitz gradient condition. Our new method just needs the locally Lipschitz gradient but still gets the rate $O(\frac{1}{k})$ of $f(x^k)-f_*$ at most. As … Read more

## Barzilai-Borwein-like rules in proximal gradient schemes for ℓ1−regularized problems

We propose a novel steplength selection rule in proximal gradient methods for minimizing the sum of a differentiable function plus an ℓ1-norm penalty term. The proposed rule modifies one of the classical Barzilai-Borwein steplength, extending analogous results obtained in the context of gradient projection methods for constrained optimization. We analyze the spectral properties of the … Read more

## A Semismooth Conjugate Gradients Method — Theoretical Analysis

In large scale applications, deterministic and stochastic variants of Cauchy’s steepest descent method are widely used for the minimization of objectives that are only piecewise smooth. In this paper we analyse a  deterministic descent method based on the generalization of rescaled conjugate gradients proposed by Philip Wolfe in 1975 for objectives that are convex. Without … Read more

## The Jordan algebraic structure of the rotated quadratic cone

In this paper, we look into the rotated quadratic cone and analyze its algebraic structure. We construct an algebra associated with this cone and show that this algebra is a Euclidean Jordan algebra (EJA) with a certain inner product. We also demonstrate some spectral and algebraic characteristics of this EJA. The rotated quadratic cone is … Read more

## A Levenberg-Marquardt Method for Nonsmooth Regularized Least Squares

 We develop a Levenberg-Marquardt method for minimizing the sum of a smooth nonlinear least-squares term $$f(x) = \frac{1}{2} \|F(x)\|_2^2$$ and a nonsmooth term $$h$$. Both $$f$$ and $$h$$ may be nonconvex. Steps are computed by minimizing the sum of a regularized linear least-squares model and a model of $$h$$ using a first-order method such … Read more

## Superiorization: The asymmetric roles of feasibility-seeking and objective function reduction

The superiorization methodology can be thought of as lying conceptually between feasibility-seeking and constrained minimization. It is not trying to solve the full-fledged constrained minimization problem composed from the modeling constraints and the chosen objective function. Rather, the task is to find a feasible point which is “superior” (in a well-defined manner) with respect to … Read more

## Orbital Crossover

Symmetry in optimization has been known to wreak havoc in optimization algorithms. Often, some of the hardest instances are highly symmetric. This is not the case in linear programming, as symmetry allows one to reduce the size of the problem, possibly dramatically, while still maintaining the same optimal objective value. This is done by aggregating … Read more

## Regularized Nonsmooth Newton Algorithms for Best Approximation

We consider the problem of finding the best approximation point from a polyhedral set, and its applications, in particular to solving large-scale linear programs. The classical projection problem has many various and many applications. We study a regularized nonsmooth Newton type solution method where the Jacobian is singular; and we compare the computational performance to … Read more