There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

다중혈관 관상동맥 환자에서 y-문합을 이용하여 양쪽 내흉동맥만을 사용한 우회술의 조기 성적

/pdf/convex-analysisnoer-san-nojin-zhan-nituite-5dl2rbmoyr.pdf

Convex Analysisの二,三の進展について

In this paper we propose new methods for solving huge-scale optimization problems. For problems of this size, even the simplest full-dimensional vector operations are very expensive. Hence, we propose to apply an optimization technique based on random partial update of decision variables. For these methods, we prove the global estimates for the rate of convergence. Surprisingly, for certain classes of objective functions, our results are better than the standard worst-case bounds for deterministic algorithms. We present constrained and unconstrained versions of the method and its accelerated variant. Our numerical test confirms a high efficiency of this technique on problems of very big size.

/pdf/efficiency-of-coordinate-descent-methods-on-huge-scale-34woq9ez3l.pdf

Efficiency of coordinate descent methods on huge-scale optimization problems

Historically, centrally computed algorithms have been the primary means of power system optimization and control. With increasing penetrations of distributed energy resources requiring optimization and control of power systems with many controllable devices, distributed algorithms have been the subject of significant research interest. This paper surveys the literature of distributed algorithms with applications to optimization and control of power systems. In particular, this paper reviews distributed algorithms for offline solution of optimal power flow (OPF) problems as well as online algorithms for real-time solution of OPF, optimal frequency control, optimal voltage control, and optimal wide-area control problems.

https://ieeexplore.ieee.org/ielaam/5165411/8075182/7990560-aam.pdf

A Survey of Distributed Optimization and Control Algorithms for Electric Power Systems

This paper proposes three strong second order cone programming (SOCP) relaxations for the AC optimal power flow (OPF) problem. These three relaxations are incomparable to each other and two of them...

https://www2.isye.gatech.edu/~xsun84/publications/OPFCycle_optonline.pdf

Strong SOCP Relaxations for the Optimal Power Flow Problem

In this paper, we examine a mixed integer linear programming reformulation for mixed integer bilinear problems where each bilinearterm involves the product of a nonnegative integer variable and a nonnegative continuous variable. This reformulation is obtained by first replacing a general integer variable with its binary expansion and then using McCormick envelopes to linearize the resulting product of continuous and binary variables. We present the convex hull of the underlying mixed integer linear set. The effectiveness of this reformulation and associated facet-defining inequalities are computationally evaluated on five classes of instances.

/pdf/solving-mixed-integer-bilinear-problems-using-milp-13ip9myd5h.pdf

Solving Mixed Integer Bilinear Problems Using MILP Formulations

It has been recently proven that the semidefinite programming (SDP) relaxation of the optimal power flow problem over radial networks is exact under technical conditions such as not including generation lower bounds or allowing load over-satisfaction. In this paper, we investigate the situation where generation lower bounds are present. We show that even for a 2-bus 1-generator system, the SDP relaxation can have all possible approximation outcomes, that is 1) SDP relaxation may be exact, 2) SDP relaxation may be inexact, or 3) SDP relaxation may be feasible while the optimal power flow (OPF) instance may be infeasible. We provide a complete characterization of when these three approximation outcomes occur and an analytical expression of the resulting optimality gap for this 2-bus system. In order to facilitate further research, we design a library of instances over radial networks in which the SDP relaxation has positive optimality gap. Finally, we propose valid inequalities and variable bound tightening techniques that significantly improve the computational performance of a global optimization solver. Our work demonstrates the need of developing efficient global optimization methods for the solution of OPF even in the simple but fundamental case of radial networks.

https://www2.isye.gatech.edu/~xsun84/publications/inexactSDP_arxiv.pdf

Inexactness of SDP Relaxation and Valid Inequalities for Optimal Power Flow

As the modern transmission control and relay technologies evolve, transmission line switching has become an important option in power system operators’ toolkits to reduce operational cost and improve system reliability. Most recent research has relied on the DC approximation of the power flow model in the optimal transmission switching problem. However, it is known that DC approximation may lead to inaccurate flow solutions and also overlook stability issues. In this paper, we focus on the optimal transmission switching problem with the full AC power flow model, abbreviated as AC optimal transmission switching (AC OTS). We propose a new exact formulation for AC OTS and its mixed-integer second-order cone programming relaxation. We improve this relaxation via several types of strong valid inequalities inspired by the recent development for the closely related AC optimal power flow problem [Kocuk et al ., “Strong SOCP relaxations for the optimal power flow problem,” Oper. Res., vol. 64, no. 6, pp. 1177–1196, 2016]. We also propose a practical algorithm to obtain high-quality feasible solutions for the AC OTS problem. Extensive computational experiments show that the proposed formulation and algorithms efficiently solve IEEE standard and congested instances and lead to significant cost benefits with provably tight bounds.

https://www2.isye.gatech.edu/~sdey30/ACSwitch.pdf

New Formulation and Strong MISOCP Relaxations for AC Optimal Transmission Switching Problem

Recently Andersen et al. [1], Borozan and Cornuejols [6] and Cornuejols and Margot [9] have characterized the extreme valid inequalities of a mixed integer set consisting of two equations with two free integer variables and non-negative continuous variables. These inequalities are either split cuts or intersection cuts derived using maximal lattice-free convex sets. In order to use these inequalities to obtain cuts from two rows of a general simplex tableau, one approach is to extend the system to include all possible non-negative integer variables (giving the two row mixed-integer infinite-group problem), and to develop lifting functions giving the coefficients of the integer variables in the corresponding inequalities. In this paper, we study the characteristics of these lifting functions. We show that there exists a unique lifting function that yields extreme inequalities when starting from a maximal lattice-free triangle with multiple integer points in the relative interior of one of its sides, or a maximal lattice-free triangle with integral vertices and one integer point in the relative interior of each side. In the other cases (maximal lattice-free triangles with one integer point in the relative interior of each side and non-integral vertices, and maximal lattice-free quadrilaterals), non-unique lifting functions may yield distinct extreme inequalities. For the latter family of triangles, we present sufficient conditions to yield an extreme inequality for the two row mixed-integer infinite-group problem.

http://www.optimization-online.org/DB_FILE/2008/06/2015.pdf

Santanu S. Dey

Papers

Strong SOCP Relaxations for the Optimal Power Flow Problem

Solving Mixed Integer Bilinear Problems Using MILP Formulations

Inexactness of SDP Relaxation and Valid Inequalities for Optimal Power Flow

New Formulation and Strong MISOCP Relaxations for AC Optimal Transmission Switching Problem

Two row mixed-integer cuts via lifting