2,239 research outputs found
A second derivative SQP method: theoretical issues
Sequential quadratic programming (SQP) methods form a class of highly efficient algorithms for solving nonlinearly constrained optimization problems. Although second derivative information may often be calculated, there is little practical theory that justifies exact-Hessian SQP methods. In particular, the resulting quadratic programming (QP) subproblems are often nonconvex, and thus finding their global solutions may be computationally nonviable. This paper presents a second-derivative SQP method based on quadratic subproblems that are either convex, and thus may be solved efficiently, or need not be solved globally. Additionally, an explicit descent-constraint is imposed on certain QP subproblems, which “guides” the iterates through areas in which nonconvexity is a concern. Global convergence of the resulting algorithm is established
Forward-backward truncated Newton methods for convex composite optimization
This paper proposes two proximal Newton-CG methods for convex nonsmooth
optimization problems in composite form. The algorithms are based on a a
reformulation of the original nonsmooth problem as the unconstrained
minimization of a continuously differentiable function, namely the
forward-backward envelope (FBE). The first algorithm is based on a standard
line search strategy, whereas the second one combines the global efficiency
estimates of the corresponding first-order methods, while achieving fast
asymptotic convergence rates. Furthermore, they are computationally attractive
since each Newton iteration requires the approximate solution of a linear
system of usually small dimension
Ghost Penalties in Nonconvex Constrained Optimization: Diminishing Stepsizes and Iteration Complexity
We consider nonconvex constrained optimization problems and propose a new
approach to the convergence analysis based on penalty functions. We make use of
classical penalty functions in an unconventional way, in that penalty functions
only enter in the theoretical analysis of convergence while the algorithm
itself is penalty-free. Based on this idea, we are able to establish several
new results, including the first general analysis for diminishing stepsize
methods in nonconvex, constrained optimization, showing convergence to
generalized stationary points, and a complexity study for SQP-type algorithms.Comment: To appear on Mathematics of Operations Researc
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