Tractable Relaxations of Composite Functions

2021 ◽  
Author(s):  
Taotao He ◽  
Mohit Tawarmalani
Keyword(s):  
Optimal Transport ◽  
Integral Formula ◽  
Transport Problem ◽  
Inner Function ◽  
Outer Function ◽  
Inner Functions ◽  
Separation Algorithm ◽  
Large Classes ◽  
Composite Functions ◽  
Factorable Programming

In this paper, we introduce new relaxations for the hypograph of composite functions assuming that the outer function is supermodular and concave extendable. Relying on a recently introduced relaxation framework, we devise a separation algorithm for the graph of the outer function over P, where P is a special polytope to capture the structure of each inner function using its finitely many bounded estimators. The separation algorithm takes [Formula: see text] time, where d is the number of inner functions and n is the number of estimators for each inner function. Consequently, we derive large classes of inequalities that tighten prevalent factorable programming relaxations. We also generalize a decomposition result and devise techniques to simultaneously separate hypographs of various supermodular, concave-extendable functions using facet-defining inequalities. Assuming that the outer function is convex in each argument, we characterize the limiting relaxation obtained with infinitely many estimators as the solution of an optimal transport problem. When the outer function is also supermodular, we obtain an explicit integral formula for this relaxation.

scholarly journals Methodology for Implementing the State Estimation in Renewable Energy Management Systems

Energies ◽  
2021 ◽  
Vol 14 (8) ◽  
pp. 2301
Author(s):  
Yun-Sung Cho ◽  
Yun-Hyuk Choi
Keyword(s):  
Renewable Energy ◽  
Power Systems ◽  
State Estimation ◽  
Measurement Errors ◽  
Large Scale ◽  
Test System ◽  
The State ◽  
Inner Function ◽  
Outer Function ◽  
Bad Data

This paper describes a methodology for implementing the state estimation and enhancing the accuracy in large-scale power systems that partially depend on variable renewable energy resources. To determine the actual states of electricity grids, including those of wind and solar power systems, the proposed state estimation method adopts a fast-decoupled weighted least square approach based on the architecture of application common database. Renewable energy modeling is considered on the basis of the point of data acquisition, the type of renewable energy, and the voltage level of the bus-connected renewable energy. Moreover, the proposed algorithm performs accurate bad data processing using inner and outer functions. The inner function is applied to the largest normalized residue method to process the bad data detection, identification and adjustment. While the outer function is analyzed whether the identified bad measurements exceed the condition of Kirchhoff’s current law. In addition, to decrease the topology and measurement errors associated with transformers, a connectivity model is proposed for transformers that use switching devices, and a transformer error processing technique is proposed using a simple heuristic method. To verify the performance of the proposed methodology, we performed comprehensive tests based on a modified IEEE 18-bus test system and a large-scale power system that utilizes renewable energy.

A discrete method for the initialization of semi-discrete optimal transport problem

2021 ◽  
Vol 212 ◽  
pp. 106608
Author(s):  
Judy Yangjun Lin ◽  
Shaoyan Guo ◽  
Longhan Xie ◽  
Ruxu Du ◽  
Gu Xu
Keyword(s):  
Optimal Transport ◽  
Transport Problem ◽  
Discrete Method

Duality for the $L^{\infty }$ optimal transport problem

2016 ◽  
Vol 369 (5) ◽  
pp. 3289-3323 ◽  
Author(s):  
E. N. Barron ◽  
M. Bocea ◽  
R. R. Jensen
Keyword(s):  
Optimal Transport ◽  
Transport Problem

scholarly journals Factorizations of outer functions and extremal problems

1996 ◽  
Vol 39 (3) ◽  
pp. 535-546 ◽  
Author(s):  
Takahiko Nakazi
Keyword(s):  
Hardy Space ◽  
Hardy Spaces ◽  
Extremal Problems ◽  
The Other ◽  
Outer Function ◽  
Inner Functions ◽  
Special Case ◽  
Class Of Functions

The author has proved that an outer function in the Hardy space H1 can be factored into a product in which one factor is strongly outer and the other is the sum of two inner functions. In an endeavor to understand better the latter factor, we introduce a class of functions containing sums of inner functions as a special case. Using it, we describe the solutions of extremal problems in the Hardy spaces Hp for 1≦p<∞.

MASS TRANSPORTATION IN GROUPS OF TYPE H

2005 ◽  
Vol 07 (04) ◽  
pp. 509-537 ◽  
Author(s):  
S. RIGOT
Keyword(s):  
Cost Function ◽  
Optimal Transport ◽  
Transport Problem ◽  
Mass Transportation ◽  
Heisenberg Groups ◽  
The One ◽  
Carathéodory Distance ◽  
The Cost ◽  
Special Case

We give a solution of the optimal transport problem in groups of type H when the cost function is the square of either the Carnot–Carathéodory or Korányi distance. This generalizes results previously proved for the Heisenberg groups. We use the same strategy that the one which was developed in that special case together with slightly refined technicalities that essentially reflect the fact that the center of the group can be of dimension larger than one. For each distance we prove existence, uniqueness and give a characterization of the optimal transport. In the case of the Carnot–Carathéodory distance we also prove that the optimal transport arises as the limit of the optimal transports in natural Riemannian approximations.

scholarly journals Optimal Mass Transport in Thin Domains

2014 ◽  
Vol 14 (1) ◽  
Author(s):  
José C. Navarro-Climent ◽  
Julio D. Rossi ◽  
Raúl C. Volpe
Keyword(s):  
Mass Transport ◽  
Euclidean Distance ◽  
Optimal Transport ◽  
Transport Problem ◽  
Thin Domains ◽  
Optimal Mass Transport

AbstractWe find the behavior of the solution of the optimal transport problem for the Euclidean distance (and its approximation by p−Laplacian problems) when the involved measures are supported in a domain that is contracted in one direction.

The Bernoulli property of inner functions

1992 ◽  
Vol 12 (2) ◽  
pp. 209-215
Author(s):  
Marcos Craizer
Keyword(s):  
Fixed Point ◽  
Unit Circle ◽  
Harmonic Measure ◽  
Bernoulli Shift ◽  
Inner Function ◽  
Inner Functions

AbstractLet f: D → D be an inner function with a fixed point p ∈ D, and f*: S1 → S1 be its extension to the unit circle. We prove in this paper that the Rohlin invertible extension of the system (f*, λp) is equivalent to a generalized Bernoulli shift, where λp is the harmonic measure associated with p.

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