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32
Snopt: An SQP Algorithm For LargeScale Constrained Optimization
, 1997
"... Sequential quadratic programming (SQP) methods have proved highly effective for solving constrained optimization problems with smooth nonlinear functions in the objective and constraints. Here we consider problems with general inequality constraints (linear and nonlinear). We assume that first deriv ..."
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Cited by 345 (19 self)
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Sequential quadratic programming (SQP) methods have proved highly effective for solving constrained optimization problems with smooth nonlinear functions in the objective and constraints. Here we consider problems with general inequality constraints (linear and nonlinear). We assume that first derivatives are available, and that the constraint gradients are sparse.
Sequential Quadratic Programming
, 1995
"... this paper we examine the underlying ideas of the SQP method and the theory that establishes it as a framework from which effective algorithms can ..."
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Cited by 117 (3 self)
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this paper we examine the underlying ideas of the SQP method and the theory that establishes it as a framework from which effective algorithms can
Feature Selection via Mathematical Programming
, 1997
"... The problem of discriminating between two finite point sets in ndimensional feature space by a separating plane that utilizes as few of the features as possible, is formulated as a mathematical program with a parametric objective function and linear constraints. The step function that appears in th ..."
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Cited by 61 (22 self)
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The problem of discriminating between two finite point sets in ndimensional feature space by a separating plane that utilizes as few of the features as possible, is formulated as a mathematical program with a parametric objective function and linear constraints. The step function that appears in the objective function can be approximated by a sigmoid or by a concave exponential on the nonnegative real line, or it can be treated exactly by considering the equivalent linear program with equilibrium constraints (LPEC). Computational tests of these three approaches on publicly available realworld databases have been carried out and compared with an adaptation of the optimal brain damage (OBD) method for reducing neural network complexity. One feature selection algorithm via concave minimization (FSV) reduced crossvalidation error on a cancer prognosis database by 35.4% while reducing problem features from 32 to 4. Feature selection is an important problem in machine learning [18, 15, 1...
On the convergence of a sequential quadratic programming method with an augmented Lagrangian line search function
 Math. Operstionsforschung und Statistik, Ser. Optimization
, 1983
"... Sequential quadratic programming (SQP) methods are widely used for solving practical optimization problems, especially in structural mechanics. The general structure of SQP methods is briefly introduced and it is shown how these methods can be adapted to distributed computing. However, SQP methods a ..."
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Cited by 34 (0 self)
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Sequential quadratic programming (SQP) methods are widely used for solving practical optimization problems, especially in structural mechanics. The general structure of SQP methods is briefly introduced and it is shown how these methods can be adapted to distributed computing. However, SQP methods are sensitive subject to errors in function and gradient evaluations. Typically they break down with an error message reporting that the line search cannot be terminated successfully. In these cases, a new nonmonotone line search is activated. In case of noisy function values, a drastic improvement of the performance is achieved compared to the version with monotone line search. Numerical results are presented for a set of more than 300 standard test examples.
Quadratically And Superlinearly Convergent Algorithms For The Solution Of Inequality Constrained Minimization Problems
, 1995
"... . In this paper some Newton and quasiNewton algorithms for the solution of inequality constrained minimization problems are considered. All the algorithms described produce sequences fx k g converging qsuperlinearly to the solution. Furthermore, under mild assumptions, a qquadratic convergence ra ..."
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Cited by 18 (6 self)
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. In this paper some Newton and quasiNewton algorithms for the solution of inequality constrained minimization problems are considered. All the algorithms described produce sequences fx k g converging qsuperlinearly to the solution. Furthermore, under mild assumptions, a qquadratic convergence rate in x is also attained. Other features of these algorithms are that the solution of linear systems of equations only is required at each iteration and that the strict complementarity assumption is never invoked. First the superlinear or quadratic convergence rate of a Newtonlike algorithm is proved. Then, a simpler version of this algorithm is studied and it is shown that it is superlinearly convergent. Finally, quasiNewton versions of the previous algorithms are considered and, provided the sequence defined by the algorithms converges, a characterization of superlinear convergence extending the result of Boggs, Tolle and Wang is given. Key Words. Inequality constrained optimization, New...
Advances in Mathematical Programming for Automated Design Integration
 KOREAN J. CHEM. ENG
, 1999
"... This paper presents a review of advances that have taken place in the mathematical programming approach to process design and synthesis. A review is first presented on the algorithms that are available for solving MINLP problems, and its most recent variant, Generalized Disjunctive Programming model ..."
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Cited by 6 (4 self)
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This paper presents a review of advances that have taken place in the mathematical programming approach to process design and synthesis. A review is first presented on the algorithms that are available for solving MINLP problems, and its most recent variant, Generalized Disjunctive Programming models. The formulation of superstructures, models and solution strategies is also discussed for the effective solution of the corresponding optimization problems. The rest of the paper is devoted to reviewing recent mathematical programming models for the synthesis of reactor networks, distillation sequences, heat exchanger networks, mass exchanger networks, utility plants, and total flowsheets. As will be seen from this review, the progress that has been achieved in this area over the last decade is very significant.
On the realization of the Wolfe conditions in reduced quasiNewton methods for equality constrained optimization
 SIAM Journal on Optimization
, 1997
"... Abstract. This paper describes a reduced quasiNewton method for solving equality constrained optimization problems. A major difficulty encountered by this type of algorithm is the design of a consistent technique for maintaining the positive definiteness of the matrices approximating the reduced He ..."
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Cited by 5 (0 self)
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Abstract. This paper describes a reduced quasiNewton method for solving equality constrained optimization problems. A major difficulty encountered by this type of algorithm is the design of a consistent technique for maintaining the positive definiteness of the matrices approximating the reduced Hessian of the Lagrangian. A new approach is proposed in this paper. The idea is to search for the next iterate along a piecewise linear path. The path is designed so that some generalized Wolfe conditions can be satisfied. These conditions allow the algorithm to sustain the positive definiteness of the matrices from iteration to iteration by a mechanism that has turned out to be efficient in unconstrained optimization.
Discrete Optimization Methods and their Role in the Integration of Planning and Scheduling
 AICHE SYMPSIUM SERIES
, 2002
"... The need for improvement in process operations, logistics and supply chain management has created a great demand for the development of optimization models for planning and scheduling. In this paper we first review the major classes of planning and scheduling models that arise in process operations, ..."
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Cited by 5 (2 self)
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The need for improvement in process operations, logistics and supply chain management has created a great demand for the development of optimization models for planning and scheduling. In this paper we first review the major classes of planning and scheduling models that arise in process operations, and establish the underlying mathematical structure of these problems. As will be shown, the nature of these models is greatly affected by the time representation (discrete or continuous), and is often dominated by discrete decisions. We then briefly review the major recent developments in mixedinteger linear and nonlinear programming, disjunctive programming and constraint programming, as well as general decomposition techniques for solving these problems. We present a general formulation for integrating planning and scheduling to illustrate the models and methods discussed in this paper.
Mathematical Programming Approaches To Machine Learning And Data Mining
, 1998
"... Machine learning problems of supervised classification, unsupervised clustering and parsimonious approximation are formulated as mathematical programs. The feature selection problem arising in the supervised classification task is effectively addressed by calculating a separating plane by minimizing ..."
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Cited by 5 (0 self)
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Machine learning problems of supervised classification, unsupervised clustering and parsimonious approximation are formulated as mathematical programs. The feature selection problem arising in the supervised classification task is effectively addressed by calculating a separating plane by minimizing separation error and the number of problem features utilized. The support vector machine approach is formulated using various norms to measure the margin of separation. The clustering problem of assigning m points in ndimensional real space to k clusters is formulated as minimizing a piecewiselinear concave function over a polyhedral set. This problem is also formulated in a novel fashion by minimizing the sum of squared distances of data points to nearest cluster planes characterizing the k clusters. The problem of obtaining a parsimonious solution to a linear system where the right hand side vector may be corrupted by noise is formulated as minimizing the system residual plus either the number of nonzero elements in the solution vector or the norm of the solution vector. The feature selection problem, the clustering problem and the parsimonious approximation problem can all be stated as the minimization of a concave function over a polyhedral region and are solved by a theoretically justifiable, fast and finite successive linearization algorithm. Numerical tests indicate the utility and efficiency of these formulations on realworld databases. In particular, the feature selection approach via concave minimization computes a separatingplane based classifier that improves upon the generalization ability of a separating plane computed without feature suppression. This approach produces ii classifiers utilizing fewer original problem features than the support vector machin...
A Piecewise LineSearch Technique for Maintaining the Positive Definiteness of the Matrices in the SQP Method
, 1997
"... Abstract. A technique for maintaining the positive definiteness of the matrices in the quasiNewton version of the SQP algorithm is proposed. In our algorithm, matrices approximating the Hessian of the augmented Lagrangian are updated. The positive definiteness of these matrices in the space tangent ..."
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Cited by 4 (2 self)
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Abstract. A technique for maintaining the positive definiteness of the matrices in the quasiNewton version of the SQP algorithm is proposed. In our algorithm, matrices approximating the Hessian of the augmented Lagrangian are updated. The positive definiteness of these matrices in the space tangent to the constraint manifold is ensured by a socalled piecewise linesearch technique, while their positive definiteness in a complementary subspace is obtained by setting the augmentation parameter. In our experiment, the combination of these two ideas leads to a new algorithm that turns out to be more robust and often improves the results obtained with other approaches.