Partition optimization with schur convex functions 5 convexity. Recognizing schur complements in nonlinear expressions is often the key step in reformulating nonlinear convex optimization problems as semidefinite programs. We apply special properties of schur convex functions to establish some inequalities for the generalized power means that include many wellknown. The schur complement is named after issai schur who used it to prove schur s lemma, although it had been used previously. Canonical problem forms ryan tibshirani convex optimization 10725. In this paper we show how to relax the problem of transduction into an sdp problem, that can then be solved by polynomial time convex optimization methods. Orf 523 lecture 9 spring 2016, princeton university instructor.
Sdp has recently been applied successfully to machine learning problems 8. Minimization of frobenius norm and schur complement. Largescale quadratic programming, activeset methods, convex and nonconvex quadratic programming, kkt systems, schur complement method, variablereduction method. Integer optimization problems are examples of nonconvex optimization problems. The consideration of coulomb friction at the contacts complicates the mathematical formulation and turns it into a nonlinear. Such a model can also be formulated as a convex, quadratic optimization problem moreau 1966. We think that convex optimization is an important enough topic that everyone who uses computational mathematics should know at least a little bit about it. Emilie virginia haynsworth was the first to call it the schur complement.
Let mbe an n nmatrix written a as 2 2 block matrix m a. Orf 523 lecture 9 spring 2016, princeton university. If is symmetric and quasi convex, then is schurconvex. This result is useful in control theo,ry where analyticity of f relates to stability of the underlying linear system. A preconditioning technique for schur complement systems arising in stochastic optimization cosmin g. A reverseschur approach to optimization with linear pde. Sergio garcia introduction to convex optimization ii june 2018 6 41. Partitionoptimization with schur convex sum objective. A generalization of the schur complement by means of the. Schur complements arise in the study of block matrix inversion i. In this paper, using the properties of schur convex function, schur geometrically convex function and schur harmonically convex function, we provide much simpler proofs of the schur convexity, schur geometric convexity and schur harmonic convexity for a composite function of. Methods for convex and general quadratic programming.
Outline lowrank matrix completion and recovery spectral methods nuclear norm minimization rip and lowrank matrix recovery phase retrieval solving random quadratic systems of equations. Named after issai schur, schur convex functions are used in the study of majorization. In this paper, we show that schur convex functions share some important properties with the ordinary convex functions. C some general classes of schurconvex functions 69 we shall show that symmetric quasi convex functions are schurconvex, but that pseudo convex functions need not be schurconvex. A reverse schur approach to optimization with linear pde constraints. A new approximation of the schur complement in preconditioners for pde. A schur complement based semiproximal admm for convex.
Statistical inference via convex optimization anatoli juditsky university grenoblealpes arkadi nemirovski georgia institute of technology. In our opinion, convex optimization is a natural next topic after advanced linear algebra topics like leastsquares, singular values, and linear programming. The goal of the current paper is to study boundedshape partitioning problems. Youll learn more about sdps next quarter if you take convex optimization. Institute of computational mathematics, johannes kepler university linz, austria. Exponential stability of primaldual gradient dynamics. Schur complement preconditioners for multiple saddle point problems of block tridiagonal form with application to optimization problems jarle sogn. A semidefinite programming method for integer convex. Every function that is convex and symmetric is also schur convex. Observe that this is always a convex optimization problem first introduced by kantorovich in the late 1930s and dantzig. The schur complement and symmetric positive semide nite. Eeacm 150 applications of convex optimization in signal. Hall thursday, march 10, 2016 when in doubt on the accuracy of these notes, please cross check with the instructors notes, on aaa.
The schur complement is a key tool in the fields of numerical analysis, statistics and matrix analysis. Completely positive programs are convex conic optimization problems which typically cannot be solved in polynomial time, so they are convex. Firstly, consider the schur complement of the block i x in matrix p 1 17, which is denoted as 1 p 1 y uu b b i x 2. Completely positive programs are convex conic optimization problems which typically cannot be solved in polynomial time, so they are convex problems that are \hard. Browse other questions tagged matrices convex optimization norm linearmatrixinequality schur complement or ask your own question. Using schur complement theorem to prove convexity of some socfunctions article pdf available in journal of nonlinear and convex analysis 3 july 2012 with 69 reads how we measure reads. Basic properties of the schur complement springerlink. In mathematics, a schurconvex function, also known as s convex, isotonic function and orderpreserving function is a function. The importance of schur complement based preconditioners is well established. Schur complements arise in many situations and appear in many important formulas and theorems. Schur complementbased substructuring of stiff multibody. The expression t xdxis called the schur complement of dtx in the matrix inequality 6. Convex optimization problems a convex optimization problem in standard form. How to prove the matrix fractional function is convex by definition.
Problem 5 can be view as a special case of the following blockseparable convex optimization problem. For convex functions, it is unclear how to take advantage of schur complement. A preconditioning technique for schur complement systems. Can you design a convex optimization algorithm that is based on schur complement. A block symmetric gaussseidel decomposition theorem for. We use the schur complement argument twice to prove lemma2. The perception is that convex problems are easy to solve to global optimality and nonconvex problems are hard. Research supported in part by national science foundation grants dms0915220 and dms18480, and by department of energy grant desc0002349. In this paper, using the properties of schurconvex function, schur geometrically convex function and schur harmonically convex function, we provide much simpler proofs of the schur convexity, schur geometric convexity and schur harmonic convexity for a composite function of the complete symmetric function.
In reverse schur co optimization, one solves the larger reverse schur complement. Optimization problems are often divided into convex and nonconvex problems. Optimization of schur convex functions xinmin zhang abstract. Inspired by the aforementioned work and with the convex quadratic conic programming in mind, we propose a schur complement based convergent semiproximal admm for solving convex programming problems, with a coupling linear equality constraint, whose objective function is the sum of two proper closed convex functions plus an arbitrary number of. A schurcomplement method for sparse quadratic programming philip e. We prove a number of discrete trace theorems, and, using these new results, show the spectral equivalence of this schur complement. How to prove the matrix fractional function is convex by. Pdf using schur complement theorem to prove convexity of. Using the schur complement result we just derived, we can write this as.
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