@techreport{NA-09/01,
  abstract = "The solution of trust-region and regularisation subproblems which arise in unconstrained optimization is considered. Building on the pioneering work of Gay, Mor� and Sorensen, methods which obtain the solution of a sequence of parametrized linear systems by factorization are used. Enhancements using high-order polynomial approximation and inverse iteration ensure that the resulting method is both globally and asymptotically at least superlinearly convergent in all cases, including in the notorious hard case. Numerical experiments validate the effectiveness of our approach. The resulting software is available as packages TRS and RQS as part of the GALAHAD optimization library, and is especially designed for large-scale problems",
  author = "H. Sue Dollar, Nicholas I. M. Gould, and Daniel P. Robinson",
  institution = "Oxford University Computing Laboratory",
  keywords = "Regularization, trust-region subproblem",
  month = "February",
  number = "NA-09/01",
  pages = "48",
  series = "NA Group technical reports",
  title = "On solving trust-region and other regularised subproblems in optimization",
  year = "2009",
}


    
      @techreport{NA-08/21,
  abstract = "In NA 08/18, we gave global convergence results for a second-derivative SQP method for minimizing the exact l1-merit function for a fixed value of the penalty parameter. To establish this result, we used the properties of the so-called Cauchy step, which was itself computed from the so-called predictor step. In addition, we allowed for the computation of a variety of (optional) SQP steps that were intended to improve the efficiency of the algorithm. Although we established global convergence of the algorithm, we did not discuss certain aspects that are critical when developing software capable of solving general optimization problems. In particular, we must have strategies for updating the penalty parameter and better techniques for defining the positive-definite matrix Bk used in computing the predictor step. In this paper we address both of these issues. We consider two techniques for defining the positive-definite matrix Bk a simple diagonal approximation and a more sophisticated limited-memory BFGS update. We also analyze a strategy for updating the penalty paramter based on approximately minimizing the l1-penalty function over a sequence of increasing values of the penalty parameter. Algorithms based on exact penalty functions have certain desirable properties. To be practical, however, these algorithms must be guaranteed to avoid the so-called Maratos effect. We show that a nonmonotone varient of our algorithm avoids this phenomenon and, therefore, results in asymptotically superlinear local convergence; this is verified by preliminary numerical results on the Hock and Shittkowski test set.",
  author = "Nicholas I. M. Gould and Daniel P. Robinson",
  institution = "Oxford University Computing Laboratory",
  month = "December",
  number = "NA-08/21",
  title = "A second derivative SQP method: Local convergence",
  year = "2008",
}


    
      @techreport{NA-08/18,
  abstract = "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.",
  author = "Nicholas I. M. Gould and Daniel P. Robinson",
  institution = "Oxford University Computing Laboratory",
  month = "November",
  number = "NA-08/18",
  title = "A second derivative SQP method: Theoretical issues",
  year = "2008",
}


    
      @techreport{NA-08/02,
  abstract = "We consider methods for regularising the least-squares solution of the linear system&nbsp;<em>Ax = b</em>. In particular, we propose iterative methods for solving large problems in which a trust-region bound<em>||x|| &le; &Delta;</em>&nbsp;is imposed on the size of the solution, and in which the least value of linear combinations of&nbsp;<em>||Ax-b||</em><sub>2</sub><sup>q</sup>&nbsp;and a regularisation term&nbsp;<em>||x||</em><sub>2</sub><sup>p</sup>&nbsp;for various&nbsp;<em>p</em>&nbsp;and&nbsp;<em>q</em>=1,2 is sought. In each case, one of more &quot;secular&quot; equations are derived, and fast Newton-like solution procedures are suggested. The resulting algorithms are available as part of the GALAHAD optimization library.",
  author = "Coralia Cartis and Nicholas I. M. Gould and Philippe L. Toint",
  institution = "Oxford University Computing Laboratory",
  month = "February",
  number = "NA-08/02",
  title = "Trust-region and other regularisations of linear least-squares problems",
  year = "2008",
}


    
      @techreport{NA-08/08,
  abstract = "Saddle-point systems arise in many applications areas, in fact in any situation where an extremum principle arises with constraints. The Stokes problem describing slow viscous flow of an incompressible fluid is a classic example coming from partial differential equations and in the area of Optimization such problems are ubiquitous. In this manuscript we show how new approaches for the solution of saddle-point systems arising in Optimization can be derived from the Bramble-Pasciak Conjugate Gradient approach widely used in PDEs and more recent generalizations thereof. In particular we derive a class of new solution methods based on the use of Preconditioned Conjugate Gradients in non-standard inner products and demonstrate how these can be understood through more standard machinery. We show connections to Constraint Preconditioning and give the results of numerical computations on a number of standard Optimization test examples.",
  author = "H. Sue Dollar and Nicholas I. M. Gould and Martin Stoll and Andrew J. Wathen",
  institution = "Oxford University Computing Laboratory",
  month = "June",
  number = "NA-08/08",
  title = "A Bramble-Pasciak-like method with applications in optimization",
  year = "2008",
}


    
      @techreport{NA-08/09,
  abstract = "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 Sl<sub>1</sub>QP 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 &quot;guides&quot; the iterates through areas in which nonconvexity is a concern. Global convergence of the resulting algorithm is established.",
  author = "Nicholas I. M. Gould and Daniel P. Robinson",
  institution = "Oxford University Computing Laboratory",
  month = "June",
  number = "NA-08/09",
  title = "A second derivative SQP method with imposed descent",
  year = "2008",
}


    
      @techreport{NA-07/20,
  abstract = "An Adaptive Cubic Overestimation (ACO) algorithm for unconstrained optimization, generalizing a method due to Nesterov & Polyak (Math. Programming 108, 2006, pp 177-205), is proposed. At each iteration of Nesterov & Polyak's approach, the global minimizer of a local cubic overestimator of the objective function is determined, and this ensures a significant improvement in the objective so long as the Hessian of the objective is Lipschitz continuous and its Lipschitz constant is available. The twin requirements of global model optimality and the availability of Lipschitz constants somewhat limit the applicability of such an approach, particularly for large-scale problems. However the promised powerful worst-case theoretical guarantees prompt us to investigate variants in which estimates of the required Lipschitz constant are refined and in which computationally-viable approximations to the global model-minimizer are sought. We show that the excellent global and local convergence properties and worst-case iteration complexity bounds obtained by Nesterov & Polyak are retained, and sometimes extended to a wider class of problems, by our ACO approach. Numerical experiments with small-scale test problems from the CUTEr set show superior performance of the ACO algorithm when compared to a trust-region implementation.",
  author = "Coralia Cartis and Nicholas I. M. Gould and Philippe L. Toint",
  institution = "Oxford University Computing Laboratory",
  month = "October",
  number = "NA-07/20",
  title = "Adaptive cubic overestimation methods for unconstrained optimization",
  year = "2007",
}