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74
Lectures on modern convex optimization
"... Mathematical Programming deals with optimization programs of the form and includes the following general areas: minimize f(x) subject to gi(x) ≤ 0, i = 1,..., m, [x ⊂ R n] 1. Modelling: methodologies for posing various applied problems as optimization programs; 2. Optimization Theory, focusing on e ..."
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Cited by 145 (6 self)
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Mathematical Programming deals with optimization programs of the form and includes the following general areas: minimize f(x) subject to gi(x) ≤ 0, i = 1,..., m, [x ⊂ R n] 1. Modelling: methodologies for posing various applied problems as optimization programs; 2. Optimization Theory, focusing on existence, uniqueness and on characterization of optimal solutions to optimization programs; 3. Optimization Methods: development and analysis of computational algorithms for various classes of optimization programs; 4. Implementation, testing and application of modelling methodologies and computational algorithms. Essentially, Mathematical Programming was born in 1948, when George Dantzig has invented Linear Programming – the class of optimization programs (P) with linear objective f(·) and
Convex Nondifferentiable Optimization: A Survey Focussed On The Analytic Center Cutting Plane Method.
, 1999
"... We present a survey of nondifferentiable optimization problems and methods with special focus on the analytic center cutting plane method. We propose a selfcontained convergence analysis, that uses the formalism of the theory of selfconcordant functions, but for the main results, we give direct pr ..."
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Cited by 70 (2 self)
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We present a survey of nondifferentiable optimization problems and methods with special focus on the analytic center cutting plane method. We propose a selfcontained convergence analysis, that uses the formalism of the theory of selfconcordant functions, but for the main results, we give direct proofs based on the properties of the logarithmic function. We also provide an in depth analysis of two extensions that are very relevant to practical problems: the case of multiple cuts and the case of deep cuts. We further examine extensions to problems including feasible sets partially described by an explicit barrier function, and to the case of nonlinear cuts. Finally, we review several implementation issues and discuss some applications.
A Primaldual InteriorPoint Method for Linear Optimization Based on a New Proximity Function
, 2002
"... In this paper we present a generic primaldual interiorpoint algorithm for linear optimization in which the search direction depends on a univariate kernel function which is also used as proximity measure in the analysis of the algorithm. We present some powerful tools for the analysis of the alg ..."
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Cited by 35 (9 self)
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In this paper we present a generic primaldual interiorpoint algorithm for linear optimization in which the search direction depends on a univariate kernel function which is also used as proximity measure in the analysis of the algorithm. We present some powerful tools for the analysis of the algorithm under the assumption that the kernel function satisfies three easy to check and mild conditions (i.e., exponential convexity, superconvexity and monotonicity of the second derivative). The approach is demonstrated by introducing a new kernel function and showing that the corresponding largeupdate algorithm improves the iteration complexity with a factor n 1 4 when compared with the classical method, which is based on the use of the logarithmic barrier function.
Multiple Cuts in the Analytic Center Cutting Plane Method
, 1998
"... We analyze the multiple cut generation scheme in the analytic center cutting plane method. We propose an optimal primal and dual updating direction when the cuts are central. The direction is optimal in the sense that it maximizes the product of the new dual slacks and of the new primal variables wi ..."
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Cited by 29 (1 self)
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We analyze the multiple cut generation scheme in the analytic center cutting plane method. We propose an optimal primal and dual updating direction when the cuts are central. The direction is optimal in the sense that it maximizes the product of the new dual slacks and of the new primal variables within the trust regions defined by Dikin's primal and dual ellipsoids. The new primal and dual directions use the variancecovariance matrix of the normals to the new cuts in the metric given by Dikin's ellipsoid. We prove that the recovery of a new analytic center from the optimal restoration direction can be done in O(p log(p + 1)) damped Newton steps, where p is the number of new cuts added by the oracle, which may vary with the iteration. The results and the proofs are independent of the specific scaling matrix primal, dual or primaldual that is used in the computations. The computation of the optimal direction uses Newton's method applied to a selfconcordant function of p variab...
Sensitivity analysis in linear programming and semidefinite programming using interiorpoint methods
 Cornell University
, 1999
"... We analyze perturbations of the righthand side and the cost parameters in linear programming (LP) and semidefinite programming (SDP). We obtain tight bounds on the norm of the perturbations that allow interiorpoint methods to recover feasible and nearoptimal solutions in a single interiorpoint i ..."
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Cited by 21 (2 self)
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We analyze perturbations of the righthand side and the cost parameters in linear programming (LP) and semidefinite programming (SDP). We obtain tight bounds on the norm of the perturbations that allow interiorpoint methods to recover feasible and nearoptimal solutions in a single interiorpoint iteration. For the unique, nondegenerate solution case in LP, we show that the bounds obtained using interiorpoint methods compare nicely with the bounds arising from the simplex method. We also present explicit bounds for SDP using the AHO, H..K..M, and NT directions.
Interiorpoint methods for optimization
, 2008
"... This article describes the current state of the art of interiorpoint methods (IPMs) for convex, conic, and general nonlinear optimization. We discuss the theory, outline the algorithms, and comment on the applicability of this class of methods, which have revolutionized the field over the last twen ..."
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Cited by 18 (0 self)
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This article describes the current state of the art of interiorpoint methods (IPMs) for convex, conic, and general nonlinear optimization. We discuss the theory, outline the algorithms, and comment on the applicability of this class of methods, which have revolutionized the field over the last twenty years.
New Complexity Analysis of the PrimalDual Newton Method for Linear Optimization
, 1998
"... We deal with the primaldual Newton method for linear optimization (LO). Nowadays, this method is the working horse in all efficient interior point algorithms for LO, and its analysis is the basic element in all polynomiality proofs of such algorithms. At present there is still a gap between the pra ..."
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Cited by 14 (8 self)
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We deal with the primaldual Newton method for linear optimization (LO). Nowadays, this method is the working horse in all efficient interior point algorithms for LO, and its analysis is the basic element in all polynomiality proofs of such algorithms. At present there is still a gap between the practical behavior of the algorithms and the theoretical performance results, in favor of the practical behavior. This is especially true for socalled largeupdate methods. We present some new analysis tools, based on a proximity measure introduced by Jansen et al., in 1994, that may help to close this gap. This proximity measure has not been used in the analysis of largeupdate method before. Our new analysis not only provides a unified way for the analysis of both largeupdate and smallupdate methods, but also improves the known iteration bounds.
On Mehrotratype predictorcorrector algorithms
, 2005
"... In this paper we discuss the polynomiality of a feasible version of Mehrotra’s predictorcorrector algorithm whose variants have been widely used in several IPM based optimization packages. A numerical example is given that shows that the adaptive choice of centering parameter and correction terms i ..."
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Cited by 13 (3 self)
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In this paper we discuss the polynomiality of a feasible version of Mehrotra’s predictorcorrector algorithm whose variants have been widely used in several IPM based optimization packages. A numerical example is given that shows that the adaptive choice of centering parameter and correction terms in this algorithm may lead to small steps being taken in order to keep the iterates in a large neighborhood of the central path, which is important to proving polynomial complexity properties of this method. Motivated by this example, we introduce a safeguard in Mehrtora’s algorithm that keeps the iterates in the prescribed neighborhood and allows us to obtain a positive lower bound on the step size. This safeguard strategy is also used when the affine scaling direction performs poorly. We prove that the safeguarded algorithm will terminate after at most O(n2 log (x0) T s0 ɛ) iteration. By modestly modifying the corrector direction, we reduce the iteration complexity to O(n log (x0) T s0 ɛ). To ensure fast asymptotic convergence of the algorithm, we changed Mehrotra’s updating scheme of the centering parameter slightly while keeping the safeguard. The new algorithms have the same order of iteration complexity as the safeguarded algorithms, but enjoy superlinear convergence as well. Numerical results using the McIPM and LIPSOL software packages are reported.
Simultaneous primaldual righthandside sensitivity analysis from a strictly complementary solution of a linear program
 SIAM J. Optim
, 2006
"... Abstract. This paper establishes theorems about the simultaneous variation of righthand sides and cost coefficients in a linear program from a strictly complementary solution. Some results are extensions of those that have been proven for varying the righthand side of the primal or the dual, but n ..."
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Cited by 13 (0 self)
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Abstract. This paper establishes theorems about the simultaneous variation of righthand sides and cost coefficients in a linear program from a strictly complementary solution. Some results are extensions of those that have been proven for varying the righthand side of the primal or the dual, but not both; other results are new. In addition, changes in the optimal partition and what that means in economic terms are related to the basisdriven approach, notably to the theory of compatibility. In addition to new theorems about this relation, the transition graph is extended to provide another visualization of the underlying economics.
Selfregular proximities and new search directions for linear and semidefinite optimization
 Mathematical Programming
, 2000
"... In this paper, we first introduce the notion of selfregular functions. Various appealing properties of selfregular functions are explored and we also discuss the relation between selfregular functions and the wellknown selfconcordant functions. Then we use such functions to define selfregular p ..."
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Cited by 12 (5 self)
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In this paper, we first introduce the notion of selfregular functions. Various appealing properties of selfregular functions are explored and we also discuss the relation between selfregular functions and the wellknown selfconcordant functions. Then we use such functions to define selfregular proximity measure for pathfollowing interior point methods for solving linear optimization (LO) problems. Any selfregular proximity measure naturally defines a primaldual search direction. In this way a new class of primaldual search directions for solving LO problems is obtained. Using the appealing properties of selfregular functions, we prove that these new largeupdate pathfollowing methods for LO enjoy a polynomial, O n q+1 2q log n iteration bound, where q ≥ 1 is the socalled barrier degree of the selfregular ε proximity measure underlying the algorithm. When q increases, this � bound approaches the √n n best known complexity bound for interior point methods, namely O log. Our unified �√n ε n analysis provides also the O log best known iteration bound of smallupdate IPMs. ε At each iteration, we need only to solve one linear system. As a byproduct of our results, we remove some limitations of the algorithms presented in [24] and improve their complexity as well. An extension of these results to semidefinite optimization (SDO) is also discussed.