# 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.

## Awards

The 2021 Beale — Orchard-Hays Prize given by MOS has been awarded to a paper published in OJMO:

Giacomo Nannicini. On the implementation of a global optimization method for mixed-variable problems. Open Journal of Mathematical Optimization, Volume 2 (2021), article  no. 1, 25 p. doi : 10.5802/ojmo.3

e-ISSN : 2777-5860

#### Short Paper - A note on the Frank–Wolfe algorithm for a class of nonconvex and nonsmooth optimization problems

Frank and Wolfe’s celebrated conditional gradient method is a well-known tool for solving smooth optimization problems for which minimizing a linear function over the feasible set is computationally cheap. However, when the objective function is nonsmooth, the method may fail to compute a stationary point. In this work, we show that the Frank–Wolfe algorithm can be employed to compute Clarke-stationary points for nonconvex and nonsmooth optimization problems consisting of minimizing upper-${C}^{1,\alpha }$ functions over convex and compact sets. Furthermore, under more restrictive assumptions, we propose a new algorithm variant with stronger stationarity guarantees, namely directional stationarity and even local optimality.

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#### Cycle-based formulations in Distance Geometry

The distance geometry problem asks to find a realization of a given simple edge-weighted graph in a Euclidean space of given dimension $K$, where the edges are realized as straight segments of lengths equal (or as close as possible) to the edge weights. The problem is often modelled as a mathematical programming formulation involving decision variables that determine the position of the vertices in the given Euclidean space. Solution algorithms are generally constructed using local or global nonlinear optimization techniques. We present a new modelling technique for this problem where, instead of deciding vertex positions, the formulations decide the length of the segments representing the edges in each cycle in the graph, projected in every dimension. We propose an exact formulation and a relaxation based on a Eulerian cycle. We then compare computational results from protein conformation instances obtained with stochastic global optimization techniques on the new cycle-based formulation and on the existing edge-based formulation. While edge-based formulations take less time to reach termination, cycle-based formulations are generally better on solution quality measures.

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