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362
Logarithmic concave measures with application to . . .
 ACTA SCIENTIARUM MATHEMATICARUM, 32 (1971), PP. 301–316.
, 1971
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An interior point algorithm for large scale nonlinear programming
 SIAM Journal on Optimization
, 1999
"... The design and implementation of a new algorithm for solving large nonlinear programming problems is described. It follows a barrier approach that employs sequential quadratic programming and trust regions to solve the subproblems occurring in the iteration. Both primal and primaldual versions of t ..."
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Cited by 78 (18 self)
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The design and implementation of a new algorithm for solving large nonlinear programming problems is described. It follows a barrier approach that employs sequential quadratic programming and trust regions to solve the subproblems occurring in the iteration. Both primal and primaldual versions of the algorithm are developed, and their performance is illustrated in a set of numerical tests. Key words: constrained optimization, interior point method, largescale optimization, nonlinear programming, primal method, primaldual method, successive quadratic programming, trust region method.
ON PROJECTED NEWTON BARRIER METHODS FOR LINEAR PROGRAMMING AND AN EQUIVALENCE TO KARMARKAR'S PROJECTIVE METHOD
, 1986
"... Interest in linear programming has been intensified recently by Karmarkar's publication in 1984 of an algorithm that is claimed to be much faster than the simplex method for practical problems. We review classical barrierfunction methods for nonlinear programming based on applying a logarithmi ..."
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Cited by 75 (8 self)
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Interest in linear programming has been intensified recently by Karmarkar's publication in 1984 of an algorithm that is claimed to be much faster than the simplex method for practical problems. We review classical barrierfunction methods for nonlinear programming based on applying a logarithmic transformation to inequality constraints. For the special case of linear programming, the transformed problem can be solved by a "projected Newton barrier" method. This method is shown to be equivalent to Karmarkar's projective method for a particular choice of the barrier parameter. We then present details of a specific barrier algorithm and its practical implementation. Numerical results are given for several nontrivial test problems, and the implications for future developments in linear programming are discussed.
Implementation of interior point methods for large scale linear programming
 Interior Point Methods in Mathematical Programming
, 1996
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The nonlinear geometry of linear programming IV. Hilbert geometry, in preparation
"... This series of papers studies a geometric structure underlying Karmarkar’s projective scaling algorithm for solving linear programming problems. A basic feature of the projective scaling algorithm is a vector field depending on the objective function which is defined on the interior of the polytope ..."
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Cited by 71 (0 self)
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This series of papers studies a geometric structure underlying Karmarkar’s projective scaling algorithm for solving linear programming problems. A basic feature of the projective scaling algorithm is a vector field depending on the objective function which is defined on the interior of the polytope of feasible solutions of the linear program. The geometric structure we study is the set of trajectories obtained by integrating this vector field, which we call Ptrajectories. In order to study Ptrajectories we also study a related vector field on the linear programming polytope, which we call the affine scaling vector field, and its associated trajectories, called Atrajectories. The affine scaling vector field is associated to another linear programming algorithm, the affine scaling algorithm. These affine and projective scaling vector fields are each defined for liner programs of a special form, called strict standard form and canonical form, respectively. This paper defines and presents basic properties of Ptrajectories and Atrajectories. It reviews the projective and affine scaling algorithms, defines the projective and affine scaling vector fields, and gives differential equations for Ptrajectories and Atrajectories. It presents Karmarkar’s interpretation of Atrajectories as steepest descent paths of the objective function 〈c, x 〉 with respect to the Riemannian _ dx
Optimal design of a CMOS opamp via geometric programming
 IEEE Transactions on ComputerAided Design
, 2001
"... We describe a new method for determining component values and transistor dimensions for CMOS operational ampli ers (opamps). We observe that a wide variety of design objectives and constraints have a special form, i.e., they are posynomial functions of the design variables. As a result the ampli er ..."
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Cited by 66 (10 self)
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We describe a new method for determining component values and transistor dimensions for CMOS operational ampli ers (opamps). We observe that a wide variety of design objectives and constraints have a special form, i.e., they are posynomial functions of the design variables. As a result the ampli er design problem can be expressed as a special form of optimization problem called geometric programming, for which very e cient global optimization methods have been developed. As a consequence we can e ciently determine globally optimal ampli er designs, or globally optimal tradeo s among competing performance measures such aspower, openloop gain, and bandwidth. Our method therefore yields completely automated synthesis of (globally) optimal CMOS ampli ers, directly from speci cations. In this paper we apply this method to a speci c, widely used operational ampli er architecture, showing in detail how to formulate the design problem as a geometric program. We compute globally optimal tradeo curves relating performance measures such as power dissipation, unitygain bandwidth, and openloop gain. We show how the method can be used to synthesize robust designs, i.e., designs guaranteed to meet the speci cations for a
Method of centers for minimizing generalized eigenvalues
 Linear Algebra Appl
, 1993
"... We consider the problem of minimizing the largest generalized eigenvalue of a pair of symmetric matrices, each of which depends affinely on the decision variables. Although this problem may appear specialized, it is in fact quite general, and includes for example all linear, quadratic, and linear fr ..."
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Cited by 63 (12 self)
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We consider the problem of minimizing the largest generalized eigenvalue of a pair of symmetric matrices, each of which depends affinely on the decision variables. Although this problem may appear specialized, it is in fact quite general, and includes for example all linear, quadratic, and linear fractional programs. Many problems arising in control theory can be cast in this form. The problem is nondifferentiable but quasiconvex, so methods such as Kelley's cuttingplane algorithm or the ellipsoid algorithm of Shor, Nemirovksy, and Yudin are guaranteed to minimize it. In this paper we describe relevant background material and a simple interior point method that solves such problems more efficiently. The algorithm is a variation on Huard's method of centers, using a selfconcordant barrier for matrix inequalities developed by Nesterov and Nemirovsky. (Nesterov and Nemirovsky have also extended their potential reduction methods to handle the same problem [NN91b].) Since the problem is quasiconvex but not convex, devising a nonheuristic stopping criterion (i.e., one that guarantees a given accuracy) is more difficult than in the convex case. We describe several nonheuristic stopping criteria that are based on the dual of a related convex problem and a new ellipsoidal approximation that is slightly sharper, in some cases, than a more general result due to Nesterov and Nemirovsky. The algorithm is demonstrated on an example: determining the quadratic Lyapunov function that optimizes a decay rate estimate for a differential inclusion.
Primaldual interior methods for nonconvex nonlinear programming
 SIAM Journal on Optimization
, 1998
"... Abstract. This paper concerns largescale general (nonconvex) nonlinear programming when first and second derivatives of the objective and constraint functions are available. A method is proposed that is based on finding an approximate solution of a sequence of unconstrained subproblems parameterize ..."
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Cited by 63 (5 self)
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Abstract. This paper concerns largescale general (nonconvex) nonlinear programming when first and second derivatives of the objective and constraint functions are available. A method is proposed that is based on finding an approximate solution of a sequence of unconstrained subproblems parameterized by a scalar parameter. The objective function of each unconstrained subproblem is an augmented penaltybarrier function that involves both primal and dual variables. Each subproblem is solved with a modified Newton method that generates search directions from a primaldual system similar to that proposed for interior methods. The augmented penaltybarrier function may be interpreted as a merit function for values of the primal and dual variables. An inertiacontrolling symmetric indefinite factorization is used to provide descent directions and directions of negative curvature for the augmented penaltybarrier merit function. A method suitable for large problems can be obtained by providing a version of this factorization that will treat large sparse indefinite systems.
A robust gradient sampling algorithm for nonsmooth, nonconvex optimization
 SIAM Journal on Optimization
"... Let f be a continuous function on R n, and suppose f is continuously differentiable on an open dense subset. Such functions arise in many applications, and very often minimizers are points at which f is not differentiable. Of particular interest is the case where f is not convex, and perhaps not eve ..."
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Cited by 61 (19 self)
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Let f be a continuous function on R n, and suppose f is continuously differentiable on an open dense subset. Such functions arise in many applications, and very often minimizers are points at which f is not differentiable. Of particular interest is the case where f is not convex, and perhaps not even locally Lipschitz, but whose gradient is easily computed where it is defined. We present a practical, robust algorithm to locally minimize such functions, based on gradient sampling. No subgradient information is required by the algorithm. When f is locally Lipschitz and has bounded level sets, and the sampling radius ǫ is fixed, we show that, with probability one, the algorithm generates a sequence with a cluster point that is Clarke ǫstationary. Furthermore, we show that if f has a unique Clarke stationary point ¯x, then the set of all cluster points generated by the algorithm converges to ¯x as ǫ is reduced to zero.
On Augmented Lagrangian methods with general lowerlevel constraints
, 2005
"... Augmented Lagrangian methods with general lowerlevel constraints are considered in the present research. These methods are useful when efficient algorithms exist for solving subproblems where the constraints are only of the lowerlevel type. Two methods of this class are introduced and analyzed. In ..."
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Cited by 55 (6 self)
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Augmented Lagrangian methods with general lowerlevel constraints are considered in the present research. These methods are useful when efficient algorithms exist for solving subproblems where the constraints are only of the lowerlevel type. Two methods of this class are introduced and analyzed. Inexact resolution of the lowerlevel constrained subproblems is considered. Global convergence is proved using the Constant Positive Linear Dependence constraint qualification. Conditions for boundedness of the penalty parameters are discussed. The reliability of the approach is tested by means of an exhaustive comparison against Lancelot. All the problems of the Cute collection are used in this comparison. Moreover, the resolution of location problems in which many constraints of the lowerlevel set are nonlinear is addressed, employing the Spectral Projected Gradient method for solving the subproblems. Problems of this type with more than 3 × 10 6 variables and 14 × 10 6 constraints are solved in this way, using moderate computer time.