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Inertiacontrolling methods for general quadratic programming
 SIAM Review
, 1991
"... Abstract. Activeset quadratic programming (QP) methods use a working set to define the search direction and multiplier estimates. In the method proposed by Fletcher in 1971, and in several subsequent mathematically equivalent methods, the working set is chosen to control the inertia of the reduced ..."
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Cited by 34 (3 self)
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Abstract. Activeset quadratic programming (QP) methods use a working set to define the search direction and multiplier estimates. In the method proposed by Fletcher in 1971, and in several subsequent mathematically equivalent methods, the working set is chosen to control the inertia of the reduced Hessian, which is never permitted to have more than one nonpositive eigenvalue. (We call such methods inertiacontrolling.) This paper presents an overview of a generic inertiacontrolling QP method, including the equations satisfied by the search direction when the reduced Hessian is positive definite, singular and indefinite. Recurrence relations are derived that define the search direction and Lagrange multiplier vector through equations related to the KarushKuhnTucker system. We also discuss connections with inertiacontrolling methods that maintain an explicit factorization of the reduced Hessian matrix. Key words. Nonconvex quadratic programming, activeset methods, Schur complement, Karush
A Computationally Efficient Feasible Sequential Quadratic Programming Algorithm
 SIAM Journal on Optimization
, 2001
"... . A sequential quadratic programming (SQP) algorithm generating feasible iterates is described and analyzed. What distinguishes this algorithm from previous feasible SQP algorithms proposed by various authors is a reduction in the amount of computation required to generate a new iterate while the pr ..."
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Cited by 28 (0 self)
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. A sequential quadratic programming (SQP) algorithm generating feasible iterates is described and analyzed. What distinguishes this algorithm from previous feasible SQP algorithms proposed by various authors is a reduction in the amount of computation required to generate a new iterate while the proposed scheme still enjoys the same global and fast local convergence properties. A preliminary implementation has been tested and some promising numerical results are reported. Key words. sequential quadratic programming, SQP, feasible iterates, feasible SQP, FSQP AMS subject classifications. 49M37, 65K05, 65K10, 90C30, 90C53 PII. S1052623498344562 1.
Convergence Criteria for Hierarchical Overlapping Coordination of Linearly Constrained Convex Problems
 Computational Optimization and Applications
, 1998
"... Decomposition of multidisciplinary engineering system design problems into smaller subproblems is desirable because it enhances robustness and understanding of the numerical results. Subproblems can be solved in parallel using the optimization technique most suitable for the underlying mathematical ..."
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Cited by 6 (5 self)
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Decomposition of multidisciplinary engineering system design problems into smaller subproblems is desirable because it enhances robustness and understanding of the numerical results. Subproblems can be solved in parallel using the optimization technique most suitable for the underlying mathematical form of the subproblem. Hierarchical overlapping coordination (HOC) is an interesting strategy for solving decomposed problems. It simultaneously uses two or more design problem decompositions, each of them associated with different partitions of the design variables and constraints. Coordination is achieved by the exchange of information between decompositions. This article presents the HOC algorithm and several new sufficient conditions for convergence of the algorithm to the optimum in the case of convex problems with linear constraints. One of these equivalent conditions involves the rank of the constraint matrix that is computationally efficienttoverify. Computational results obtained b...
Block structured quadratic programming for the direct multiple shooting method for optimal control
 Optimization Methods and Software
, 2011
"... Abstract. In this contribution we address the efficient solution of optimal control problems of dynamic processes with many controls. Such problems arise, e.g., from the outer convexification of integer control decisions. We treat this optimal control problem class using the direct multiple shooting ..."
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Cited by 4 (4 self)
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Abstract. In this contribution we address the efficient solution of optimal control problems of dynamic processes with many controls. Such problems arise, e.g., from the outer convexification of integer control decisions. We treat this optimal control problem class using the direct multiple shooting method to discretize the optimal control problem. The resulting nonlinear problems are solved using sequential quadratic programming methods. We review the classical condensing algorithm that preprocesses the large but sparse quadratic programs to obtain small but dense ones. We show that this approach leaves room for improvement when applied in conjunction with outer convexification. To this end, we present a new complementary condensing algorithm for quadratic programs with many controls. This algorithm is based on a hybrid null–space range–space approach to exploit the block sparse structure of the quadratic programs that is due to direct multiple shooting. An assessment of the theoretical run time complexity reveals significant advantages of the proposed algorithm. We give a detailed account on the required number of floating point operations, depending on the process dimensions. Finally we demonstrate the merit of the new complementary condensing approach by comparing the behavior of both methods for a vehicle control problem in which the integer gear decision is convexified. 1.
An NE/SQP Method for the Bounded Nonlinear Complementarity Problem
 Preprint MCSP5080495, Argonne National Laboratory, Argonne
, 1995
"... NE/SQP is a recent algorithm that has proven quite effective for solving the pure and mixed forms of the nonlinear complementarity problem (NCP). NE/SQP is robust in the sense that its directionfinding subproblems are always solvable; in addition, the convergence rate of this method is Qquadratic. ..."
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Cited by 2 (2 self)
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NE/SQP is a recent algorithm that has proven quite effective for solving the pure and mixed forms of the nonlinear complementarity problem (NCP). NE/SQP is robust in the sense that its directionfinding subproblems are always solvable; in addition, the convergence rate of this method is Qquadratic. In this paper we consider a generalized version of NE/SQP proposed by Pang and Qi, that is suitable for the bounded NCP. We extend their work by demonstrating a stronger convergence result and then test a proposed method on several numerical problems. Key Words: Nonlinear complementarity problem, mathematical programming, sequential quadratic programming. 1 Introduction In a recent paper [18] Pang and Qi presented a general algorithm to solve various mathematical programs that can be formulated as systems of nonsmooth equations. Various convergence results were shown for several general nonsmooth formulations. One specific nonsmooth system considered was the upper bounded nonlinear compl...
A SECOND DERIVATIVE SQP METHOD WITH IMPOSED DESCENT
, 2008
"... 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 exactHessian SQP methods. In particul ..."
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Cited by 2 (0 self)
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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 exactHessian 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 secondderivative Sℓ1QP 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.
Complementary Condensing for the Direct Multiple Shooting Method
"... Summary. In this contribution we address the efficient solution of optimal control problems of dynamic processes with many controls. Such problems typically arise from the convexification of integer control decisions. We treat this problem class using the direct multiple shooting method to discretiz ..."
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Cited by 1 (1 self)
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Summary. In this contribution we address the efficient solution of optimal control problems of dynamic processes with many controls. Such problems typically arise from the convexification of integer control decisions. We treat this problem class using the direct multiple shooting method to discretize the optimal control problem. The resulting nonlinear problems are solved using an SQP method. Concerning the solution of the quadratic subproblems we present a factorization of the QP’s KKT system, based on a combined null–space range–space approach exploiting the problem’s block sparse structure. We demonstrate the merit of this approach for a vehicle control problem in which the integer gear decision is convexified. 1
User’s Guide for SQOPT Version 7: Software for LargeScale Linear and Quadratic Programming ∗
, 2008
"... SQOPT is a software package for minimizing a convex quadratic function subject to both equality and inequality constraints. SQOPT may also be used for linear programming and for finding a feasible point for a set of linear equalities and inequalities. SQOPT uses a twophase, activeset, reduceHessi ..."
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Cited by 1 (0 self)
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SQOPT is a software package for minimizing a convex quadratic function subject to both equality and inequality constraints. SQOPT may also be used for linear programming and for finding a feasible point for a set of linear equalities and inequalities. SQOPT uses a twophase, activeset, reduceHessian method. It is most efficient on problems with relatively few degrees of freedom (for example, if only some of the variables appear in the quadratic term, or the number of active constraints and bounds is nearly as large as the number of variables). However, unlike previous versions of SQOPT, there is no limit on the number of degrees of freedom. SQOPT is primarily intended for large linear and quadratic problems with sparse constraint matrices. A quadratic term 1 2 xTHx in the objective function is represented by a user subroutine that returns the product Hx for a given vector x. SQOPT uses stable numerical methods throughout and includes a reliable basis package (for maintaining sparse LU factors of the basis matrix), a practical antidegeneracy procedure, scaling, and elastic bounds on any number of constraints and variables. SQOPT is part of the SNOPT package for largescale nonlinearly constrained optimization. The source code is reentrant and suitable for any machine with a Fortran compiler (or the f2c translator and a C compiler). SQOPT may be called from a driver program in Fortran, C, or Matlab. It can also be used as a standalone package, reading data in the MPS format used by commercial mathematical programming systems.
A SECOND DERIVATIVE SQP METHOD: THEORETICAL ISSUES
, 2008
"... 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 exactHessian SQP methods. In particul ..."
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Cited by 1 (1 self)
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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 exactHessian 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 secondderivative 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 descentconstraint 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.