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62
SQP methods for largescale nonlinear programming
, 1999
"... We compare and contrast a number of recent sequential quadratic programming (SQP) methods that have been proposed for the solution of largescale nonlinear programming problems. Both linesearch and trustregion approaches are considered, as are the implications of interiorpoint and quadratic progr ..."
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Cited by 9 (0 self)
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We compare and contrast a number of recent sequential quadratic programming (SQP) methods that have been proposed for the solution of largescale nonlinear programming problems. Both linesearch and trustregion approaches are considered, as are the implications of interiorpoint and quadratic programming methods.
Solving Problems with Semidefinite and Related Constraints Using InteriorPoint Methods for Nonlinear Programming. this volume
, 2002
"... Abstract. In this paper, we describe how to reformulate a problem that has secondorder cone and/or semidefiniteness constraints in order to solve it using a generalpurpose interiorpoint algorithm for nonlinear programming. The resulting problems are smooth and convex, and numerical results from t ..."
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Cited by 8 (1 self)
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Abstract. In this paper, we describe how to reformulate a problem that has secondorder cone and/or semidefiniteness constraints in order to solve it using a generalpurpose interiorpoint algorithm for nonlinear programming. The resulting problems are smooth and convex, and numerical results from the DIMACS Implementation Challenge problems and SDPLib are provided. 1.
Assessing the Potential of Interior Methods for Nonlinear Optimization
, 2002
"... A series of numerical experiments with interior point (LOQO, KNITRO) and activeset sequential quadratic programming (SNOPT, filterSQP) codes are reported and analyzed. The tests were performed with small, mediumsize and moderately large problems, and are examined by problem classes. Detailed obser ..."
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Cited by 7 (1 self)
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A series of numerical experiments with interior point (LOQO, KNITRO) and activeset sequential quadratic programming (SNOPT, filterSQP) codes are reported and analyzed. The tests were performed with small, mediumsize and moderately large problems, and are examined by problem classes. Detailed observations on the performance of the codes, and several suggestions on how to improve them are presented. Overall, interior methods appear to be strong competitors of activeset SQP methods, but all codes show much room for improvement. 1
Jr., The Symmetric Eigenvalue Complementarity Problem
 Mathematics of Computation
"... Abstract. This paper is devoted to the Eigenvalue Complementarity Problem (EiCP) with symmetric real matrices. This problem is equivalent to finding a stationary point of a differentiable optimization program involving the Rayleigh quotient on a simplex [22]. We discuss a logarithmic function and a ..."
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Cited by 7 (3 self)
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Abstract. This paper is devoted to the Eigenvalue Complementarity Problem (EiCP) with symmetric real matrices. This problem is equivalent to finding a stationary point of a differentiable optimization program involving the Rayleigh quotient on a simplex [22]. We discuss a logarithmic function and a quadratic programming formulation to find a complementarity eigenvalue by computing a stationary point of an appropriate merit function on a special convex set. A variant of the spectral projected gradient algorithm with a specially designed line search is introduced to solve the EiCP. Computational experience shows that the application of this algorithm to the logarithmic function formulation is a quite efficient way to find a solution to the symmetric EiCP. 1.
A twosided relaxation scheme for mathematical programs with equilibrium constraints
 SIAM J. Optim
, 2005
"... Abstract. We propose a relaxation scheme for mathematical programs with equilibrium constraints (MPECs). In contrast to previous approaches, our relaxation is twosided: both the complementarity and the nonnegativity constraints are relaxed. The proposed relaxation update rule guarantees (under cert ..."
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Cited by 7 (0 self)
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Abstract. We propose a relaxation scheme for mathematical programs with equilibrium constraints (MPECs). In contrast to previous approaches, our relaxation is twosided: both the complementarity and the nonnegativity constraints are relaxed. The proposed relaxation update rule guarantees (under certain conditions) that the sequence of relaxed subproblems will maintain a strictly feasible interior—even in the limit. We show how the relaxation scheme can be used in combination with a standard interiorpoint method to achieve superlinear convergence. Numerical results on the MacMPEC test problem set demonstrate the fast local convergence properties of the approach. Key words. nonlinear programming, mathematical programs with equilibrium constraints, complementarity constraints, constrained minimization, interiorpoint methods, primaldual methods,
The N k Problem in Power Grids: New Models, Formulations and Numerical Experiments (extended version) 1
, 2008
"... Given a power grid modeled by a network together with equations describing the power flows, power generation and consumption, and the laws of physics, the socalled N − k problem asks whether there exists a set of k or fewer arcs whose removal will cause the system to fail. The case where k is small ..."
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Cited by 7 (1 self)
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Given a power grid modeled by a network together with equations describing the power flows, power generation and consumption, and the laws of physics, the socalled N − k problem asks whether there exists a set of k or fewer arcs whose removal will cause the system to fail. The case where k is small is of practical interest. We present theoretical and computational results involving a mixedinteger model and a continuous nonlinear model related to this question. 1
An interiorpoint method for MPECs based on strictly feasible relaxations
 Preprint ANL/MCSP11500404, Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, IL
, 2004
"... Abstract. An interiorpoint method for solving mathematical programs with equilibrium constraints (MPECs) is proposed. At each iteration of the algorithm, a single primaldual step is computed from each subproblem of a sequence. Each subproblem is defined as a relaxation of the MPEC with a nonempty ..."
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Cited by 6 (0 self)
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Abstract. An interiorpoint method for solving mathematical programs with equilibrium constraints (MPECs) is proposed. At each iteration of the algorithm, a single primaldual step is computed from each subproblem of a sequence. Each subproblem is defined as a relaxation of the MPEC with a nonempty strictly feasible region. In contrast to previous approaches, the proposed relaxation scheme preserves the nonempty strict feasibility of each subproblem even in the limit. Local and superlinear convergence of the algorithm is proved even with a less restrictive strict complementarity condition than the standard one. Moreover, mechanisms for inducing global convergence in practice are proposed. Numerical results on the MacMPEC test problem set demonstrate the fastlocal convergence properties of the algorithm. Key words. nonlinear programming, mathematical programs with equilibrium constraints, constrained minimization, interiorpoint methods, primaldual methods, barrier methods
The Penalty Interior Point Method fails to converge for mathematical programs with equilibrium constraints
 University of Dundee
, 2002
"... Equilibrium equations in the form of complementarity conditions often appear as constraints in optimization problems. Problems of this type are commonly referred to as mathematical programs with complementarity constraints (MPCCs). A popular method for solving MPCCs is the penalty interiorpoint alg ..."
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Cited by 6 (1 self)
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Equilibrium equations in the form of complementarity conditions often appear as constraints in optimization problems. Problems of this type are commonly referred to as mathematical programs with complementarity constraints (MPCCs). A popular method for solving MPCCs is the penalty interiorpoint algorithm (PIPA). This paper presents a small example for which PIPA converges to a nonstationary point, providing a counterexample to the established theory. The reasons for this adverse behavior are discussed.
Infeasible ConstraintReduced InteriorPoint Methods for Linear Optimization ∗
, 2010
"... Constraintreduction schemes have been proposed for the solution by interiorpoint methods of linear programs with many more inequality constraints than variables in standard dual form. Such schemes have been shown to be provably convergent and highly efficient in practice. A critical requirement of ..."
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Cited by 5 (3 self)
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Constraintreduction schemes have been proposed for the solution by interiorpoint methods of linear programs with many more inequality constraints than variables in standard dual form. Such schemes have been shown to be provably convergent and highly efficient in practice. A critical requirement of these schemes is the availability of an initial dualfeasible point. In this paper, building on a general framework (which encompasses several previously proposed approaches) for dualfeasible constraintreduced interiorpoint optimization, for which we prove convergence to a single point of the sequence of dual iterates, we propose a framework for “infeasible ” constraintreduced interiorpoint optimization. Central to this framework is an exact (ℓ1 or ℓ∞) penalty function scheme endowed with a mechanism for iterative adjustment of the penalty parameter, which aims at yielding, after a finite number of iterations, a value that guarantees feasibility (for the original problem) of the minimizers. Finiteness of the sequence of penalty parameter adjustments is proved under mild assumptions for all algorithms that fit within the framework, including “infeasible ” extensions of a “dual ” algorithm proposed in the early 1990s and of two recently proposed “primaldual ” algorithms. One of the latter two, a constraintreduced variant of Mehrotra’s PredictorCorrector algorithm, is then more specifically considered: further convergence results are proved, and numerical results are reported that demonstrate that the approach is of practical interest.
A Comparative Study of LargeScale Nonlinear Optimization Algorithms
, 2001
"... In recent years, much work has been done on implementing a variety of algorithms in nonlinear programming software. In this paper, we analyze the performance of several stateof theart optimization codes on largescale nonlinear optimization problems. Extensive numerical results are presented on di ..."
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Cited by 5 (0 self)
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In recent years, much work has been done on implementing a variety of algorithms in nonlinear programming software. In this paper, we analyze the performance of several stateof theart optimization codes on largescale nonlinear optimization problems. Extensive numerical results are presented on di#erent classes of problems, and features of each code that make it e#cient or ine#cient for each class are examined. 1.