scholarly journals Oracle-Based Primal-Dual Algorithms for Packing and Covering Semidefinite Programs

2021 ◽  
pp. 47-63
Author(s):  
Khaled Elbassioni ◽  
Kazuhisa Makino
Keyword(s):  
Combinatorial Optimization ◽  
Optimization Problems ◽  
Logarithmic Potential ◽  
Efficient Algorithms ◽  
Semidefinite Programs ◽  
Packing And Covering ◽  
Combinatorial Optimization Problems ◽  
Primal Dual ◽  
Potential Methods

AbstractPacking and covering semidefinite programs (SDPs) appear in natural relaxations of many combinatorial optimization problems as well as a number of other applications. Recently, several techniques have been proposed that utilize the particular structure of this class of problems in order to obtain more efficient algorithms than those offered by general SDP solvers. For certain applications, it may be necessary to deal with SDPs with a very large number of (e.g., exponentially or even infinitely many) constraints. In this chapter, we give an overview of some of the techniques that can be used to solve this class of problems, focusing on multiplicative weight updates and logarithmic-potential methods.

scholarly journals Quantum SDP-Solvers: Better upper and lower bounds

Quantum ◽  
2020 ◽  
Vol 4 ◽  
pp. 230 ◽  
Author(s):  
Joran van Apeldoorn ◽  
András Gilyén ◽  
Sander Gribling ◽  
Ronald de Wolf
Keyword(s):  
Combinatorial Optimization ◽  
Lower Bounds ◽  
Optimization Problems ◽  
Quantum Algorithms ◽  
Upper And Lower Bounds ◽  
Smooth Functions ◽  
Worst Case ◽  
Semidefinite Programs ◽  
New Techniques ◽  
Combinatorial Optimization Problems

Brandão and Svore \cite{brandao2016QSDPSpeedup} recently gave quantum algorithms for approximately solving semidefinite programs, which in some regimes are faster than the best-possible classical algorithms in terms of the dimension n of the problem and the number m of constraints, but worse in terms of various other parameters. In this paper we improve their algorithms in several ways, getting better dependence on those other parameters. To this end we develop new techniques for quantum algorithms, for instance a general way to efficiently implement smooth functions of sparse Hamiltonians, and a generalized minimum-finding procedure.We also show limits on this approach to quantum SDP-solvers, for instance for combinatorial optimization problems that have a lot of symmetry. Finally, we prove some general lower bounds showing that in the worst case, the complexity of every quantum LP-solver (and hence also SDP-solver) has to scale linearly with mn when m≈n, which is the same as classical.

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