# 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

#### 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.

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#### Estimating Shape Parameters of Piecewise Linear-Quadratic Problems

Piecewise Linear-Quadratic (PLQ) penalties are widely used to develop models in statistical inference, signal processing, and machine learning. Common examples of PLQ penalties include least squares, Huber, Vapnik, 1-norm, and their asymmetric generalizations. Properties of these estimators depend on the choice of penalty and its shape parameters, such as degree of asymmetry for the quantile loss, and transition point between linear and quadratic pieces for the Huber function.

In this paper, we develop a statistical framework that can help the modeler to automatically tune shape parameters once the shape of the penalty has been chosen. The choice of the parameter is informed by the basic notion that each PLQ penalty should correspond to a true statistical density. The normalization constant inherent in this requirement helps to inform the optimization over shape parameters, giving a joint optimization problem over these as well as primary parameters of interest. A second contribution is to consider optimization methods for these joint problems. We show that basic first-order methods can be immediately brought to bear, and design specialized extensions of interior point (IP) methods for PLQ problems that can quickly and efficiently solve the joint problem. Synthetic problems and larger-scale practical examples illustrate the utility of the approach. Code for the new IP method is implemented using the Julia language (https://github.com/UW-AMO/shape-parameters/tree/ojmo).

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