One algorithm for branch and bound method for solving concave optimization problem

Program Linier Integer Murni merupakan optimisasi kombinatorial yang tidak mudah untuk diselesaikan secara efisien. Metode yang sering digunakan untuk menyelesaikan Program Linier Integer Murni diantaranya adalah metode merative, yang merupakan salah satunya metode Branch and Bound. Metode ini menggunakan hasil dari metode simpleks yang belum bernilai integer sehingga dilakukan pencabangan dan batasan terhadap variabel x_j yang bernilai pecahan terbesar. Metode Branch and Bound dapat menyelesaikan masalah optimisasi suatu produk, tetapi membutuhkan waktu yang lebih lama dalam proses perhitungannya dikarenakan dalam setiap tahap perhitungan harus dicari nilai dari batas atas dan batas bawah yang ditentuan berdasarkan suatubatasandankriteria tertentu. Pure Integer Linear Program is combinatorial optimization that is not easy to solve efficiently. The method that is often used to complete the Pure Integer Linear Program is the merative method, which is one of the Branch and Bound methods. This method uses the results of the simplex method that is not yet an integer value so that the branching and limitation of the x_j variable is the largest fraction. The Branch and Bound method can solve the optimization problem of a product, but requires a longer time in the calculation process because in each calculation phase, a value must be sought from the upper and lower limits determined based on the boundary and certain criteria.

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A geometric branch and bound method for robust maximization of convex functions

Journal of Global Optimization ◽

10.1007/s10898-021-01038-7 ◽

2021 ◽

Author(s):

Fengqiao Luo ◽

Sanjay Mehrotra

Keyword(s):

Branch And Bound ◽

Convex Functions ◽

Branch And Bound Method

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Revisiting Modified Greedy Algorithm for Monotone Submodular Maximization with a Knapsack Constraint

Proceedings of the ACM on Measurement and Analysis of Computing Systems ◽

10.1145/3447386 ◽

2021 ◽

Vol 5 (1) ◽

pp. 1-22

Author(s):

Jing Tang ◽

Xueyan Tang ◽

Andrew Lim ◽

Kai Han ◽

Chongshou Li ◽

...

Keyword(s):

Approximation Algorithms ◽

Greedy Algorithm ◽

Branch And Bound ◽

Upper Bound ◽

Optimization Problem ◽

Approximation Factor ◽

Real World Application ◽

Efficiency Of Algorithms ◽

Knapsack Constraint ◽

Submodular Maximization

Monotone submodular maximization with a knapsack constraint is NP-hard. Various approximation algorithms have been devised to address this optimization problem. In this paper, we revisit the widely known modified greedy algorithm. First, we show that this algorithm can achieve an approximation factor of 0.405, which significantly improves the known factors of 0.357 given by Wolsey and (1-1/e)/2\approx 0.316 given by Khuller et al. More importantly, our analysis closes a gap in Khuller et al.'s proof for the extensively mentioned approximation factor of (1-1/\sqrte )\approx 0.393 in the literature to clarify a long-standing misconception on this issue. Second, we enhance the modified greedy algorithm to derive a data-dependent upper bound on the optimum. We empirically demonstrate the tightness of our upper bound with a real-world application. The bound enables us to obtain a data-dependent ratio typically much higher than 0.405 between the solution value of the modified greedy algorithm and the optimum. It can also be used to significantly improve the efficiency of algorithms such as branch and bound.

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