# Open Journal of Mathematical Optimization

This is the home page of the Open Journal of Mathematical Optimization, an electronic journal of computer science and mathematics owned by its Editorial Board.

The Open Journal of Mathematical Optimization (OJMO) publishes original and high-quality articles dealing with every aspect of mathematical optimization, ranging from numerical and computational aspects to the theoretical questions related to mathematical optimization problems. The topics covered by the journal are classified into four areas:

1. Continuous Optimization
2. Discrete Optimization
3. Optimization under Uncertainty
4. Computational aspects and applications

The journal publishes high-quality articles in open access free of charge, meaning that neither the authors nor the readers have to pay to access the content of the published papers, thus adhering to the principles of Fair Open Access. The journal supports open data and open code whenever possible and authors are strongly encouraged to submit code and data sets along with their manuscript.

### Memberships and Indexing

e-ISSN : 2777-5860

#### A note on approximate accelerated forward-backward methods with absolute and relative errors, and possibly strongly convex objectives

In this short note, we provide a simple version of an accelerated forward-backward method (a.k.a. Nesterov’s accelerated proximal gradient method) possibly relying on approximate proximal operators and allowing to exploit strong convexity of the objective function. The method supports both relative and absolute errors, and its behavior is illustrated on a set of standard numerical experiments.

Using the same developments, we further provide a version of the accelerated proximal hybrid extragradient method of [21] possibly exploiting strong convexity of the objective function.

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#### Efficient optimization of the Held–Karp lower bound

Given a weighted undirected graph $G=\left(V,E\right)$, the Held–Karp lower bound for the Traveling Salesman Problem (TSP) is obtained by selecting an arbitrary vertex $\overline{p}\in V$, by computing a minimum cost tree spanning $V\setminus \left\{\overline{p}\right\}$ and adding two minimum cost edges adjacent to $\overline{p}$. In general, different selections of vertex $\overline{p}$ provide different lower bounds. In this paper it is shown that the selection of vertex $\overline{p}$ can be optimized, to obtain the largest possible Held–Karp lower bound, with the same worst-case computational time complexity required to compute a single minimum spanning tree. Although motivated by the optimization of the Held–Karp lower bound for the TSP, the algorithm solves a more general problem, allowing for the efficient pre-computation of alternative minimum spanning trees in weighted graphs where any vertex can be deleted.

Available online: