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Advances in convex optimization: Conic programming
 In Proceedings of International Congress of Mathematicians
, 2007
"... Abstract. During the last two decades, major developments in convex optimization were focusing on conic programming, primarily, on linear, conic quadratic and semidefinite optimization. Conic programming allows to reveal rich structure which usually is possessed by a convex program and to exploit ..."
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Abstract. During the last two decades, major developments in convex optimization were focusing on conic programming, primarily, on linear, conic quadratic and semidefinite optimization. Conic programming allows to reveal rich structure which usually is possessed by a convex program and to exploit this structure in order to process the program efficiently. In the paper, we overview the major components of the resulting theory (conic duality and primaldual interior point polynomial time algorithms), outline the extremely rich “expressive abilities ” of conic quadratic and semidefinite programming and discuss a number of instructive applications.
Support recovery via weighted maximumcontrast subagging
, 2013
"... Abstract. In this paper, we study finite sample properties of subagging for nonsmooth estimation and model selection in sparse and largescale regression settings where both the number of parameters and the number of samples can be extremely large. This setup is very different from highdimensional ..."
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Abstract. In this paper, we study finite sample properties of subagging for nonsmooth estimation and model selection in sparse and largescale regression settings where both the number of parameters and the number of samples can be extremely large. This setup is very different from highdimensional regression and is such that Lasso estimator might be inappropriate for computational, rather than statistical reasons. We show that subagging of Lasso estimators results in discontinuous estimated support set and is never able to recover sparsity set when at least one of aggregated estimators has probability of support recovery strictly less than 1. Therefore, we propose its randomized and smoothed alternative, which we call weighted maximumcontrast subagging. We develop theory in support of the claim that proposed method has tight error control over both false positives and false negatives, regardless of the size of a dataset. Unlike existing methods, it allows for oraclelike properties, even in cases of nonoraclelike properties of aggregated estimators. Furthermore, we design an adaptive procedure for selecting tuning parameters and appropriate optimal weighting scheme. Finally, we validate our theoretical findings through extensive simulation study and analysis of a part of the millionsongchallenge dataset.
Reduced Vertex Set Result for Interval Semidefinite Optimization Problems
, 2008
"... In this paper we propose a reduced vertex result for the robust solution of uncertain semidefinite optimization problems subject to interval uncertainty. If the number of decision variables is m and the size of the coefficient matrices in the linear matrix inequality constraints is n × n, a direct ..."
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In this paper we propose a reduced vertex result for the robust solution of uncertain semidefinite optimization problems subject to interval uncertainty. If the number of decision variables is m and the size of the coefficient matrices in the linear matrix inequality constraints is n × n, a direct vertex approach would require satisfaction of 2 n(m+1)(n+1)/2 vertex constraints: a huge number, even for small values of n and m. The conditions derived here are instead based on the introduction of m slack variables and a subset of vertex coefficient matrices of cardinality 2 n−1, thus reducing the problem to a practically manageable size, at least for small n. A similar size reduction is also obtained for a class of problems with affinely dependent interval uncertainty.
SFB 823 Adaptive grid semidefinite programming for finding optimal designs Discussion Paper
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DUAL METHODS FOR THE NUMERICAL SOLUTION OF THE UNIVARIATE POWER MOMENT PROBLEM
, 2003
"... The purpose of this paper is twofold. First to present a brief survey of some of the basic results related to the univariate moment problem, including Prékopa's dual approach for solving the discrete moment problem. Second we propose a new method for solving the continuous power moment problem ..."
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The purpose of this paper is twofold. First to present a brief survey of some of the basic results related to the univariate moment problem, including Prékopa's dual approach for solving the discrete moment problem. Second we propose a new method for solving the continuous power moment problem when some higher order divided differences of the objective function are nonnegative. The proposed method combines Prékopa's dual approach for solving the discrete moment problem with a cuttingplane type procedure for solving linear semiinfinite programming problems.
Probabilistic and Setbased Model Invalidation and Estimation using LMIs
, 2013
"... Probabilistic and setbased methods are two approaches for model (in)validation, parameter and state estimation. Both classes of methods use different types of data, i.e. deterministic or probabilistic data, which allow different statements and applications. Ideally, however, all available data sho ..."
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Probabilistic and setbased methods are two approaches for model (in)validation, parameter and state estimation. Both classes of methods use different types of data, i.e. deterministic or probabilistic data, which allow different statements and applications. Ideally, however, all available data should be used in estimation and model invalidation methods. This paper presents an estimation and model (in)validation framework combining setbased and probabilistically uncertain data for polynomial continuoustime systems. In particular, uncertain data on the moments and the support is used without the need to make explicit assumptions on the type of probability densities. The paper derives pointwiseintime outer approximations of the moments of the probability densities associated with the states and parameters of the system. These approximations can be interpreted as guaranteed confidence intervals for the moment estimates. Furthermore, guaranteed bounds on the probability masses on subsets are derived and allow an estimation of the unknown probability densities. To calculate the estimates, the dynamics of the probability densities of the state trajectories are found by occupation measures of the nonlinear dynamics. This allows the construction of an infinitedimensional linear program which incorporates the set and momentbased data. This linear program is relaxed by a hierarchy of LMI problems providing, as shown elsewhere, an almost uniformly convergent sequence of outer approximations of the estimated sets. The approach is demonstrated with numerical examples. 1