In this paper, we propose an infeasible arc-search interior-point algorithm for solving nonlinear programming problems. Most algorithms based on interior-point methods are categorized as line search since they compute a next iterate on a straight line determined by a search direction which approximates the central path. The proposed arc-search interior-point algorithm uses an arc for the approximation. We discuss convergence properties of the proposed algorithm. We also conduct numerical experiments on the CUTEst benchmark problems and compare the performance of the proposed arc-search algorithm with that of a line-search algorithm. Numerical results indicate that the proposed arc-search algorithm reaches the optimal solution using fewer iterations but longer times than a line-search algorithm. A modification that leads to a faster arc-search algorithm is also discussed.

This research was partially supported by JSPS KAKENHI (Grant Number: 18K11176).

Authors and Affiliations

1. Department of Mathematical and Computing Science, Tokyo Institute of Technology, Tokyo, Japan

   Makoto Yamashita & Einosuke Iida

2. US NRC, Office of Research, 21 Church Street, Rockville, MD, USA

   Yaguang Yang

Corresponding author

Correspondence toMakoto Yamashita.

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Appendix: 1: Derivatives

In this section, we give notation related to derivatives. The Hessian matrix of\(f : {\mathbb {R}}^{n} \to {\mathbb {R}}\) is

The Jacobian for\(h : {\mathbb {R}}^{n} \to {\mathbb {R}}^{m}\) is

The Jacobian for\(g : {\mathbb {R}}^{n} \to {\mathbb {R}}^{p}\) is

For the right-hand side of (8), we use

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Appendix: 2: The largest step angle

In this section, we give analytical forms to compute the largest\(\alpha _{w_{i}}\) and\(\alpha _{s_{i}}\) for eachi in (12). For simplicity, here, we drop the indexi and the iteration numberk; for example,\({w_{i}^{k}}\) is simply written asw. For (11a), we should have

or equivalently,

We split this computation into seven cases by the signs of\(\dot {w}\) and\(\ddot {w}\).

Case 1

(\(\dot {w}=0\)and\(\ddot {w}\ne 0\))

If\(\ddot {w} \ge -(1-\delta ) w\), thenw(α) ≥δw holds for\(\alpha \in [0, \frac {\pi }{2}]\). If\(\ddot {w} \le -(1-\delta )w < 0\), to meet (28), we must have\(\cos \limits (\alpha ) \ge \frac {w-\delta w +\ddot {w}}{\ddot {w}} \ge 0\), or,\(\alpha \le \cos \limits ^{-1}\left (\frac {w-\delta w +\ddot {w}} {\ddot {w}} \right )\). Therefore,

Case 2

(\(\ddot {w}=0\)and\(\dot {w}\ne 0\))

If\(\dot {w} \le (1 - \delta ) w\), thenw(α) ≥δw holds for any\(\alpha \in [0, \frac {\pi }{2}]\). If\(\dot {w} \ge (1 - \delta ) w > 0\), to meet (28), we must have\(\sin \limits (\alpha ) \le \frac {w-\delta w }{\dot {w}}\), or\(\alpha \le \sin \limits ^{-1}\left (\frac {w-\delta w } {\dot {w}} \right )\). Therefore,

Case 3

(\(\dot {w}>0\)and\(\ddot {w}>0\))

Let\(\beta = \sin \limits ^{-1} \left (\frac {\ddot {w} } {\sqrt {\dot {w}^{2}+\ddot {w}^{2}}} \right )\). We can express\(\dot {w}=\sqrt {\dot {w}^{2}+\ddot {w}^{2}}\cos \limits (\beta )\) and\(\ddot {w}=\sqrt {\dot {w}^{2}+\ddot {w}^{2}}\sin \limits (\beta )\). Then, (28) can be rewritten as

If\(\ddot {w} + w- \delta w \ge \sqrt {\dot {w}^{2}+\ddot {w}^{2}}\), thenw(α) ≥δw holds for any\(\alpha \in [0, \frac {\pi }{2}]\). If\(\ddot {w} + w- \delta w \le \sqrt {\dot {w}^{2}+\ddot {w}^{2}}\), to meet (29), we must have\(\sin \limits (\alpha + \beta ) \le \frac {w-\delta w + \ddot {w}} {\sqrt {\dot {w}^{2}+\ddot {w}^{2}}}\), or\(\alpha + \beta \le \sin \limits ^{-1}\left (\frac {w-\delta w + \ddot {w} } {\sqrt {\dot {w}^{2}+\ddot {w}^{2}}} \right )\). Therefore,

Case 4

(\(\dot {w}>0\)and\(\ddot {w}<0\))

Let\(\beta = \sin \limits ^{-1} \left (\frac {-\ddot {w} } {\sqrt {\dot {w}^{2}+\ddot {w}^{2}}} \right )\). We can express\(\dot {w}=\sqrt {\dot {w}^{2}+\ddot {w}^{2}}\cos \limits (\beta )\) and\(\ddot {w}=-\sqrt {\dot {w}^{2}+\ddot {w}^{2}}\sin \limits (\beta )\). Then, (28) can be rewritten as

If\(\ddot {w} + w- \delta w \ge \sqrt {\dot {w}^{2}+\ddot {w}^{2}}\), thenw(α) ≥δw holds for any\(\alpha \in [0, \frac {\pi }{2}]\). If\(\ddot {w} +w- \delta w \le \sqrt {\dot {w}^{2}+\ddot {w}^{2}}\), to meet (30), we must have\(\sin \limits (\alpha - \beta ) \le \frac {w-\delta w + \ddot {w}} {\sqrt {\dot {w}^{2}+\ddot {w}^{2}}}\), or\(\alpha - \beta \le \sin \limits ^{-1}\left (\frac {w-\delta w + \ddot {w} } {\sqrt {\dot {w}^{2}+\ddot {w}^{2}}} \right )\). Therefore,

Case 5

(\(\dot {w}<0\)and\(\ddot {w}<0\))

Let\( \beta = \sin \limits ^{-1} \left (\frac {-\ddot {w} } {\sqrt {\dot {w}^{2}+\ddot {w}^{2}}} \right )\). We can express\(\dot {w}=-\sqrt {\dot {w}^{2}+\ddot {w}^{2}}\cos \limits (\beta )\) and\(\ddot {w}=-\sqrt {\dot {w}^{2}+\ddot {w}^{2}}\sin \limits (\beta )\). Then, (28) can be rewritten as

If\(\ddot {w} +(w- \delta w) \ge 0\), thenw(α) ≥δw holds for any\(\alpha \in [0, \frac {\pi }{2}]\). If\(\ddot {w} +(w- \delta w) \le 0\), to meet (31), we must have\(\sin \limits (\alpha + \beta ) \ge \frac {-(w-\delta w + \ddot {w})} {\sqrt {\dot {w}^{2}+\ddot {w}^{2}}}\), or\(\alpha + \beta \le \pi - \sin \limits ^{-1} \left (\frac {-(w-\delta w + \ddot {w})}{\sqrt {\dot {w}^{2}+\ddot {w}^{2}}} \right )\). Therefore,

Case 6

(\(\dot {w}<0\)and\(\ddot {w}>0\))

Clearly, (28) always holds for any\(\alpha \in [0, \frac {\pi }{2}]\). Therefore, we can take

Case 7

(\(\dot {w}=0\)and\(\ddot {w}=0\))

Clearly, (28) always holds for any\(\alpha \in [0, \frac {\pi }{2}]\). Therefore, we can take

Similar analysis can be performed for (11b), then similar analytical forms are derived forα_s.

Appendix: 3: Details on numerical results

Tables 1 and 2 report the objective value, the numbers of iterations, and the computation time (in seconds) of the proposed arc-search algorithm (Algorithm 1) and the line-search algorithm (Algorithm 2) for QCQP and Other type problems. The symbol “Unattained” indicates that the algorithms stopped prematurely, mainly because of the numerical errors. We excluded the problems that all the three algorithms (Algorithm 1, Algorithm 1 with\(\ddot {\ddot {v}}\), and Algorithm 2) stopped with “Unattained.”

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