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A Newton-CG based barrier-augmented Lagrangian method for general nonconvex conic optimization

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Abstract

In this paper we consider finding an approximate second-order stationary point (SOSP) of general nonconvex conic optimization that minimizes a twice differentiable function subject to nonlinear equality constraints and also a convex conic constraint. In particular, we propose a Newton-conjugate gradient (Newton-CG) based barrier-augmented Lagrangian method for finding an approximate SOSP of this problem. Under some mild assumptions, we show that our method enjoys a total inner iteration complexity of \({\widetilde{{{\,\mathrm{\mathcal {O}}\,}}}}(\epsilon ^{-11/2})\) and an operation complexity of \({\widetilde{{{\,\mathrm{\mathcal {O}}\,}}}}(\epsilon ^{-11/2}\min \{n,\epsilon ^{-5/4}\})\) for finding an \((\epsilon ,\sqrt{\epsilon })\)-SOSP of general nonconvex conic optimization with high probability. Moreover, under a constraint qualification, these complexity bounds are improved to \({\widetilde{{{\,\mathrm{\mathcal {O}}\,}}}}(\epsilon ^{-7/2})\) and \({\widetilde{{{\,\mathrm{\mathcal {O}}\,}}}}(\epsilon ^{-7/2}\min \{n,\epsilon ^{-3/4}\})\), respectively. To the best of our knowledge, this is the first study on the complexity of finding an approximate SOSP of general nonconvex conic optimization. Preliminary numerical results are presented to demonstrate superiority of the proposed method over first-order methods in terms of solution quality.

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Algorithm 1
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Data availibility

The codes for generating the random data and implementing the algorithms in the numerical section are available from the first author upon request.

Notes

  1. In fact, a total inner iteration complexity of \({\widetilde{{{\,\mathrm{\mathcal {O}}\,}}}}(\epsilon ^{-7})\) and an operation complexity \({\widetilde{{{\,\mathrm{\mathcal {O}}\,}}}}(\epsilon ^{-7}\min \{n,\epsilon ^{-1}\})\) were established in [70] for finding an \((\epsilon ,\epsilon )\)-SOSP of problem (1) with high probability; see [70, Theorem 4(ii), Corollary 3(ii), Theorem 5] Nevertheless, they can be easily modified to obtain the aforementioned complexity for finding an \((\epsilon ,\sqrt{\epsilon })\)-SOSP of (1) with high probability.

  2. The operation complexity of the barrier method [43] is measured by the amount of fundamental operations consisting of matrix–vector products, matrix multiplications, Cholesky factorizations, and backward or forward substitutions to a triangular linear system.

  3. The arithmetic complexity of Cholesky factorizations for a positive definite matrix is \({{\,\mathrm{\mathcal {O}}\,}}(n^3)\) in general, while the arithmetic complexity of matrix–vector products and backward or forward substitutions is at most \({{\,\mathrm{\mathcal {O}}\,}}(n^2)\), where n is the number of rows of the matrix.

  4. It shall be mentioned that the total numbers of Cholesky factorizations are only \({\widetilde{{{\,\mathrm{\mathcal {O}}\,}}}}(\epsilon ^{-7/2})\) and \({\widetilde{{{\,\mathrm{\mathcal {O}}\,}}}}(\epsilon ^{-11/2})\) respectively for the case where constraint qualification holds or not. See Subsections 5.3 and 5.4 for details.

  5. The \(\lambda ^{k+1}\) is also called a safeguarded Lagrangian multiplier, which has been used in the literature for designing some AL methods (e.g., see [3, 13, 44, 47]). It has been shown to enjoy many practical and theoretical advantages (e.g., see [13]).

  6. The vectorization of a matrix is the column vector obtained by stacking the columns of the matrix on top of one another.

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Authors and Affiliations

  1. Department of Computer Science and Engineering, University of Minnesota, Minneapolis, USA

    Chuan He

  2. Department of Mathematics, Linköping University, Linköping, Sweden

    Chuan He

  3. Department of Computer Science, University of Maryland, College Park, USA

    Heng Huang

  4. Department of Industrial and Systems Engineering, University of Minnesota, Minneapolis, USA

    Zhaosong Lu

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Appendices

Appendix

A capped conjugate gradient method

Algorithm 3
figure c

A capped conjugate gradient method

Full size image

In this part we present the capped CG method proposed in [60, Algorithm 1] for solving a possibly indefinite linear system (15). As briefly discussed in Sect. 4, the capped CG method finds either an approximate solution to (15) or a sufficiently negative curvature direction of the associated matrix H. More details about this method can be found in [60, Section 3.1].

The following theorem presents the iteration complexity of Algorithm 3, whose proof can be found in [44, Theorem A.1], and thus omitted here.

Theorem A.1

(iteration complexity of Algorithm 3) Consider applying Algorithm 3 with the optional input \(U=0\) to the linear system (15) with \(g\ne 0\), \(\varepsilon >0\), and H being an \(n\times n\) symmetric matrix. Then the number of iterations of Algorithm 3 is \({\widetilde{{{\,\mathrm{\mathcal {O}}\,}}}}(\min \{n,\sqrt{\Vert H\Vert /\varepsilon }\})\).

A randomized Lanczos based minimum eigenvalue oracle

Algorithm 4
figure d

A randomized Lanczos based minimum eigenvalue oracle

Full size image

In this part we present the randomized Lanczos method proposed in [60, Section 3.2], which can be used as a minimum eigenvalue oracle for Algorithm 1. As mentioned in Sect. 4, this oracle either outputs a sufficiently negative curvature direction of H or certifies that H is nearly positive semidefinite with high probability. More details about it can be found in [60, Section 3.2].

The following theorem justifies that Algorithm 4 is a suitable minimum eigenvalue oracle for Algorithm 1. Its proof is identical to that of [60, Lemma 2] and thus omitted.

Theorem B.1

[Iteration complexity of Algorithm 4] Consider Algorithm 4 with tolerance \(\varepsilon >0\), probability parameter \(\delta \in (0,1)\), and symmetric matrix \(H\in {{\,\mathrm{\mathbb {R}}\,}}^{n\times n}\) as its input. Then it either finds a sufficiently negative curvature direction v satisfying \(v^THv\le -\varepsilon /2\) and \(\Vert v\Vert =1\) or certifies that \(\lambda _{\min }(H)\ge -\varepsilon \) holds with probability at least \(1-\delta \) in at most \(N(\varepsilon ,\delta )\) iterations, where \(N(\varepsilon ,\delta )\) is defined in (111).

Notice that generally, computing \(\Vert H\Vert \) in Algorithm 4 may not be cheap when n is large. Nevertheless, \(\Vert H\Vert \) can be efficiently estimated via a randomization scheme with high confidence (e.g., see the discussion in [60, Appendix B3]).

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He, C., Huang, H. & Lu, Z. A Newton-CG based barrier-augmented Lagrangian method for general nonconvex conic optimization. Comput Optim Appl 89, 843–894 (2024). https://doi.org/10.1007/s10589-024-00603-6

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