21 research outputs found
Comment on "Geometry of the quantum set on no-signaling faces"
In Ref. [1] the authors claim that the Almost Quantum set of correlations
cannot reproduce two points on the boundary of the quantum set of correlations.
This claim is incorrect. The underlying issue is that the associated SDP is not
strictly feasible, which makes the numerical solvers give unreliable answers.
We give analytical proofs that both points are indeed reproduced by Almost
Quantum.Comment: 5 pages, 2 Mathematica notebook
ЗАДАЧИ ЛИНЕЙНОГО ПОЛУОПРЕДЕЛЕННОГО ПРОГРАММИРОВАНИЯ: РЕГУЛЯРИЗАЦИЯ И ДВОЙСТВЕННЫЕ ФОРМУЛИРОВКИ В СТРОГОЙ ФОРМЕ
Regularisation consists in reducing a given optimisation problem to an equivalent form where certain regularity conditions,
which guarantee the strong duality, are fulfilled. In this paper, for linear problems of semidefinite programming
(SDP), we propose a regularisation procedure which is based on the concept of an immobile index set and its
properties. This procedure is described in the form of a finite algorithm which converts any linear semidefinite problem
to a form that satisfies the Slater condition. Using the properties of the immobile indices and the described regularisation
procedure, we obtained new dual SDP problems in implicit and explicit forms. It is proven that for the constructed dual
problems and the original problem the strong duality property holds true.Регуляризация задачи оптимизации состоит в ее сведении к эквивалентной задаче, удовлетворяющей условиям
регулярности, которые гарантируют выполнение соотношений двойственности в строгой форме. В настоящей
статье для линейных задач полуопределенного программирования предлагается процедура регуляризации, основанная на понятии неподвижных индексов и их свойствах. Эта процедура описана в виде алгоритма, который за
конечное число шагов преобразует любую задачу линейного полубесконечного программирования в эквивалентную задачу, удовлетворяющую условию Слейтера. В результате использования свойств неподвижных индексов
и предложенной процедуры регуляризации получены новые двойственные задачи полубесконечного программирования в явной и неявной формах. Доказано, что для этих двойственных задач и исходной задачи соотношения
двойственности выполняются в строгой форме.publishe
A Strict Complementarity Approach to Error Bound and Sensitivity of Solution of Conic Programs
In this paper, we provide an elementary, geometric, and unified framework to
analyze conic programs that we call the strict complementarity approach. This
framework allows us to establish error bounds and quantify the sensitivity of
the solution. The framework uses three classical ideas from convex geometry and
linear algebra: linear regularity of convex sets, facial reduction, and
orthogonal decomposition. We show how to use this framework to derive error
bounds for linear programming (LP), second order cone programming (SOCP), and
semidefinite programming (SDP).Comment: 19 pages, 2 figure
Analytic Formulas for Alternating Projection Sequences for the Positive Semidefinite Cone and an Application to Convergence Analysis
We find analytic formulas for the alternating projection method for the cone
of positive semidefinite matrices and an affine subspace. More
precisely, we find recursive relations on parameters representing a sequence
constructed by the alternating projection method. By applying the formulas, we
analyze the alternating projection method in detail and show that the upper
bound given by the singularity degree is actually tight when the alternating
projection method is applied to and a -plane whose
intersection is a singleton and has singularity degree .Comment: 26 page