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Constructing Boosting Algorithms from SVMs: An Application to Oneclass Classification
, 2002
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Barrier Boosting
"... Boosting algorithms like AdaBoost and ArcGV are iterative strategies to minimize a constrained objective function, equivalent to Barrier algorithms. ..."
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Cited by 19 (7 self)
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Boosting algorithms like AdaBoost and ArcGV are iterative strategies to minimize a constrained objective function, equivalent to Barrier algorithms.
Penalty/barrier multiplier algorithm for semidefinite programming
 Optimization Methods and Software
"... We present a generalization of the Penalty/Barrier Multiplier algorithm for the semidefinite programming, based on a matrix form of Lagrange multipliers. Our approach allows to use among others logarithmic, shifted logarithmic, exponential and a very effective quadraticlogarithmic penalty/barrier f ..."
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Cited by 19 (7 self)
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We present a generalization of the Penalty/Barrier Multiplier algorithm for the semidefinite programming, based on a matrix form of Lagrange multipliers. Our approach allows to use among others logarithmic, shifted logarithmic, exponential and a very effective quadraticlogarithmic penalty/barrier functions. We present dual analysis of the method, based on its correspondence to a proximal point algorithm with nonquadratic distancelike function. We give computationally tractable dual bounds, which are produced by the Legendre transformation of the penalty function. Numerical results for largescale problems from robust control, robust truss topology design and free material design demonstrate high efficiency of the algorithm. 1
Fast SDP Algorithms for Constraint Satisfaction Problems
"... The class of constraint satisfactions problems (CSPs) captures many fundamental combinatorial optimization problems such as Max Cut, Max qCut, Unique Games, and Max kSat. Recently, Raghavendra (STOC‘08) identified a simple semidefinite programming relaxation that gives the best possible approximat ..."
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Cited by 8 (3 self)
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The class of constraint satisfactions problems (CSPs) captures many fundamental combinatorial optimization problems such as Max Cut, Max qCut, Unique Games, and Max kSat. Recently, Raghavendra (STOC‘08) identified a simple semidefinite programming relaxation that gives the best possible approximation for any CSP, assuming the Unique Games Conjecture. Raghavendra and Steurer (FOCS‘09) showed that, independent of the truth of the Unique Games Conjecture, the integrality gap of this relaxation cannot be improved even by adding a large class of valid inequalities. We present an algorithm that finds an approximately optimal solution to this relaxation in nearlinear time. Combining this algorithm with a rounding scheme of Raghavendra and Steurer (FOCS‘09) leads to an approximation algorithm for any CSP that runs in nearlinear time and has an approximation guarantee that matches the integrality gap, which is optimal assuming the Unique Games Conjecture.
SVM and Boosting: One Class
"... We show via an equivalence of mathematical programs that a Support Vector (SV) algorithm can be translated into an equivalent boostinglike algorithm and vice versa. We exemplify this translation procedure for a new algorithm oneclass Leveraging starting from the oneclass Support Vector Machine ..."
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Cited by 6 (1 self)
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We show via an equivalence of mathematical programs that a Support Vector (SV) algorithm can be translated into an equivalent boostinglike algorithm and vice versa. We exemplify this translation procedure for a new algorithm oneclass Leveraging starting from the oneclass Support Vector Machines (1SVM) . This is a first step towards unsupervised learning in a Boosting framework.
Central paths in semidefinite programming, generalized proximalpoint method and Cauchy trajectories in Riemannian manifolds
 J. Optim. Theory Appl
"... The relationships among central path in the context of semidefinite programming, generalized proximal point method and Cauchy trajectory in Riemannian manifolds is studied in this paper. First it is proved that the central path associated to the general function is well defined. The convergence and ..."
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The relationships among central path in the context of semidefinite programming, generalized proximal point method and Cauchy trajectory in Riemannian manifolds is studied in this paper. First it is proved that the central path associated to the general function is well defined. The convergence and characterization of its limit point is established for functions satisfying a certain continuous property. Also, the generalized proximal point method is considered, and it is proved that the corresponding generated sequence is contained in the central path. As a consequence, both converge to the same point. Finally, it is proved that the central path coincides with the Cauchy trajectory in the Riemannian manifold.
Generalized Proximal Point Algorithms
"... In this paper, we address the problem of minimizing a lower semicontinuous function f over R . More precisely, we study the convergence properties of a general proximal point scheme of the form where is a distancelike bivariate function. The analysis is performed without convexity nor ..."
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In this paper, we address the problem of minimizing a lower semicontinuous function f over R . More precisely, we study the convergence properties of a general proximal point scheme of the form where is a distancelike bivariate function. The analysis is performed without convexity nor dierentiability assumptions. Applications of this work are to extensions of EMtype algorithms in statistical estimation. We also extend the study to the problem of minimizing f over the nonnegative orthant via entropiclike proximal methods. In particular, convergence of entropictype methods is obtained with greatly reduced regularity assumptions in comparison to previous published works.
On the Convergence of Central Path and Generalized Proximal Point Method for Symmetric Cone Linear Programming
"... Abstract: In this paper, we consider the symmetric cone linear programming(SCLP), by using the Jordanalgebraic technique, we extend the generalized proximal point method in linear programming and semidefinite programming to the SCLP. Under some reasonable conditions, we obtain the convergence of pr ..."
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Abstract: In this paper, we consider the symmetric cone linear programming(SCLP), by using the Jordanalgebraic technique, we extend the generalized proximal point method in linear programming and semidefinite programming to the SCLP. Under some reasonable conditions, we obtain the convergence of primal central paths associated to the symmetric cone distance function.
Proximal Point Methods for Functions Involving Lojasiewicz, Quasiconvex and Convex Properties on Hadamard Manifolds
"... This paper extends the full convergence of the proximal point method with Riemannian, SemiBregman and Bregman distances to solve minimization problems on Hadamard manifolds. For the unconstrained problem, under the assumptions that the optimal set is nonempty and the objective function is continuou ..."
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This paper extends the full convergence of the proximal point method with Riemannian, SemiBregman and Bregman distances to solve minimization problems on Hadamard manifolds. For the unconstrained problem, under the assumptions that the optimal set is nonempty and the objective function is continuous and either quasiconvex or satisfies a generalized Lojasiewicz property, we prove the full convergence of the sequence generated by the proximal point method with Riemannian distances to certain generalized critical point of the problem. For the constrained case, under the same assumption on the optimal set and the quasiconvexity or convexity of the objective function, we study two methods. One of them is the proximal method with semiBregman distance, obtaining that any cluster point of the sequence is an optimal solution. The other one, is the proximal method with Bregman distance where we obtain the global convergence of the method to an optimal solution of the problem. In particular, our work recovers some interesting optimization problems, for example, convex and quasiconvex minimization problems in IR n, semidefinite problems (SDP), second order cone problems (SOCP), in the same way that, extends the applications of the proximal point methods for solving constrained minimization problems with nonconvex objective functions in Euclidian spaces when the objective function is convex or quasiconvex on the manifold.