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39
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 146 (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
Semidefinite Programming and Combinatorial Optimization
 DOC. MATH. J. DMV
, 1998
"... We describe a few applications of semide nite programming in combinatorial optimization. ..."
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Cited by 106 (1 self)
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We describe a few applications of semide nite programming in combinatorial optimization.
On maximization of quadratic form over intersection of ellipsoids with common center
, 1999
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New Results on Quadratic Minimization
, 2001
"... In this paper we present several new results on minimizing an indefinite quadratic function under quadratic/linear constraints. The emphasis is placed on the case where the constraints are two quadratic inequalities. This formulation is known as the extended trust region subproblem and the computati ..."
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Cited by 65 (8 self)
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In this paper we present several new results on minimizing an indefinite quadratic function under quadratic/linear constraints. The emphasis is placed on the case where the constraints are two quadratic inequalities. This formulation is known as the extended trust region subproblem and the computational complexity of this problem is still unknown. We consider several interesting cases related to this problem and show that for those cases the corresponding SDP relaxation admits no gap with the true optimal value, and consequently we obtain polynomial time procedures for solving those special cases of quadratic optimization. For the extended trust region subproblem itself, we introduce a parameterized problem and prove the existence of a trajectory which will lead to an optimal solution. Combining with a result obtained in the first part of the paper, we propose a polynomialtime solution procedure for the extended trust region subproblem arising from solving nonlinear programs with a single equality constraint.
On Approximating Complex Quadratic Optimization Problems Via Semidefinite Programming Relaxations
 Mathematical Programming, Series B
, 2007
"... Abstract. In this paper we study semidefinite programming (SDP) models for a class of discrete and continuous quadratic optimization problems in the complex Hermitian form. These problems capture a class of well–known combinatorial optimization problems, as well as problems in control theory. For i ..."
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Cited by 30 (4 self)
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Abstract. In this paper we study semidefinite programming (SDP) models for a class of discrete and continuous quadratic optimization problems in the complex Hermitian form. These problems capture a class of well–known combinatorial optimization problems, as well as problems in control theory. For instance, they include Max–3–Cut with arbitrary edge weights (i.e. some of the edge weights might be negative). We present a generic algorithm and a unified analysis of the SDP relaxations which allow us to obtain good approximation guarantees for our models. Specifically, we give an (k sin(pi/k))2/(4pi)–approximation algorithm for the discrete problem where the decision variables are k– ary and the objective matrix is positive semidefinite. To the best of our knowledge, this is the first known approximation result for this family of problems. For the continuous problem where the objective matrix is positive semidefinite, we obtain the well–known pi/4 result due to [2], and independently, [12]. However, our techniques simplify their analyses and provide a unified framework for treating these problems. In addition, we show for the first time that the integrality gap of the SDP relaxation is precisely pi/4. We also show that the unified analysis can be used to obtain an O(1 / log n)–approximation algorithm for the continuous problem in the case where the objective matrix is not positive semidefinite. 1
On nonapproximability for quadratic programs
 IN 46TH ANNUAL SYMPOSIUM ON FOUNDATIONS OF COMPUTER SCIENCE
, 2005
"... This paper studies the computational complexity of the following type of quadratic programs: given an arbitrary matrix whose diagonal elements are zero, find x ∈ {−1, 1} n that maximizes x T Mx. This problem recently attracted attention due to its application in various clustering settings, as well ..."
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Cited by 28 (4 self)
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This paper studies the computational complexity of the following type of quadratic programs: given an arbitrary matrix whose diagonal elements are zero, find x ∈ {−1, 1} n that maximizes x T Mx. This problem recently attracted attention due to its application in various clustering settings, as well as an intriguing connection to the famous Grothendieck inequality. It is approximable to within a factor of O(log n), and known to be NPhard to approximate within any factor better than 13/11 − ɛ for all ɛ> 0. We show that it is quasiNPhard to approximate to a factor better than O(log γ n) for some γ> 0. The integrality gap of the natural semidefinite relaxation for this problem is known as the Grothendieck constant of the complete graph, and known to be Θ(log n). The proof of this fact was nonconstructive, and did not yield an explicit problem instance where this integrality gap is achieved. Our techniques yield an explicit instance for which the integrality gap is log n Ω ( log log n), essentially answering one of the open problems of Alon et al. [AMMN].
HighDimensional Shape Fitting in Linear Time
 Discrete Comput. Geom
, 2002
"... The radius of a kdimensional at F with respect to P , denoted by RD(F ; P ), is de ned to be max p2P dist(F ; p), where dist(F ; p) denotes the Euclidean distance between p and its projection onto F . The kat radius of P , which we denote by R k (P ), is the minimum, over all kdimensional at ..."
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Cited by 25 (7 self)
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The radius of a kdimensional at F with respect to P , denoted by RD(F ; P ), is de ned to be max p2P dist(F ; p), where dist(F ; p) denotes the Euclidean distance between p and its projection onto F . The kat radius of P , which we denote by R k (P ), is the minimum, over all kdimensional ats F , of RD(F ; P ). We consider the problem of computing R k (P ) for a given set of points P .
SDP gaps and UGChardness for MaxCutGain
, 2008
"... Given a graph with maximum cut of (fractional) size c, the Goemans–Williamson semidefinite programming (SDP)based algorithm is guaranteed to find a cut of size at least.878 · c. However this guarantee becomes trivial when c is near 1/2, since making random cuts guarantees a cut of size 1/2 (i.e., ..."
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Cited by 25 (4 self)
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Given a graph with maximum cut of (fractional) size c, the Goemans–Williamson semidefinite programming (SDP)based algorithm is guaranteed to find a cut of size at least.878 · c. However this guarantee becomes trivial when c is near 1/2, since making random cuts guarantees a cut of size 1/2 (i.e., half of all edges). A few years ago, Charikar and Wirth (analyzing an algorithm of Feige and Langberg) showed that given a graph with maximum cut 1/2 + ε, one can find a cut of size 1/2 + Ω(ε/log(1/ε)). The main contribution of our paper is twofold: 1. We give a natural and explicit 1/2 + ε vs. 1/2 + O(ε/log(1/ε)) integrality gap for the MaxCut SDP based on Euclidean space with the Gaussian probability distribution. This shows that the SDProunding algorithm of CharikarWirth is essentially best possible. 2. We show how this SDP gap can be translated into a Long Code test with the same parameters. This implies that beating the CharikarWirth guarantee with any efficient algorithm is NPhard, assuming the Unique Games Conjecture (UGC). This result essentially settles the asymptotic approximability of MaxCut, assuming UGC. Building on the first contribution, we show how “randomness reduction ” on related SDP gaps for the QuadraticProgramming problem lets us make the Ω(log(1/ε)) gap as large as Ω(logn) for nvertex graphs. In addition to optimally answering an open question
Nonconvex mixedinteger nonlinear programming: A survey
 Surveys in Operations Research and Management Science
, 2012
"... A wide range of problems arising in practical applications can be formulated as MixedInteger Nonlinear Programs (MINLPs). For the case in which the objective and constraint functions are convex, some quite effective exact and heuristic algorithms are available. When nonconvexities are present, how ..."
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Cited by 21 (0 self)
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A wide range of problems arising in practical applications can be formulated as MixedInteger Nonlinear Programs (MINLPs). For the case in which the objective and constraint functions are convex, some quite effective exact and heuristic algorithms are available. When nonconvexities are present, however, things become much more difficult, since then even the continuous relaxation is a global optimisation problem. We survey the literature on nonconvex MINLP, discussing applications, algorithms and software. Special attention is paid to the case in which the objective and constraint functions are quadratic. Key Words: mixedinteger nonlinear programming, global optimisation, quadratic programming, polynomial optimisation.