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
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
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
To explore the limits of a stochastic gradient method, it may be useful to consider an example consisting of an infinite number of quadratic functions. In this context, it is appropriate to determine the expected value and the covariance matrix of the stochastic noise, i.e. the difference of the true gradient and the approximated gradient … Read more
We first provide an inner-approximation hierarchy described by a sum-of-squares (SOS) constraint for the copositive (COP) cone over a general symmetric cone. The hierarchy is a generalization of that proposed by Parrilo (2000) for the usual COP cone (over a nonnegative orthant). We also discuss its dual. Second, we characterize the COP cone over a … Read more
This note considers the momentum method by Polyak and the accelerated gradient method by Nesterov, both without line search but with fixed step length applied to strictly convex quadratic functions assuming that exact gradients are used and appropriate upper and lower bounds for the extreme eigenvalues of the Hessian matrix are known. Simple 2-d-examples show … Read more
We consider a generalization of two-stage decision problems in which the second-stage decision may be a function of a predictive signal but cannot adapt fully to the realized uncertainty. We will show how such problems can be learned from sample data by considering a family of regularized sample average formulations. Furthermore, our regularized data-driven formulations … Read more
We establish the weak convergence of inertial Krasnoselskii-Mann iterations towards a common fixed point of a family of quasi-nonexpansive operators, along with worst case estimates for the rate at which the residuals vanish. Strong and linear convergence are obtained in the quasi-contractive setting. In both cases, we highlight the relationship with the non-inertial case, and … Read more
A new projective exact penalty function method is proposed for the equivalent reduction of constrained optimization problems to nonsmooth unconstrained ones. In the method, the original objective function is extended to infeasible points by summing its value at the projection of an infeasible point on the feasible set with the distance to the projection. The … Read more
Under mild assumptions stochastic gradient methods asymptotically achieve an optimal rate of convergence if the arithmetic mean of all iterates is returned as an approximate optimal solution. However, in the absence of stochastic noise, the arithmetic mean of all iterates converges considerably slower to the optimal solution than the iterates themselves. And also in the … Read more