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Large-scale quasi-Newton trust-region methods with low-dimensional linear equality constraints


We propose two limited-memory BFGS (L-BFGS) trust-region methods for large-scale optimization with linear equality constraints. The methods are intended for problems where the number of equality constraints is small. By exploiting the structure of the quasi-Newton compact representation, both proposed methods solve the trust-region subproblems nearly exactly, even for large problems. We derive theoretical global convergence results of the proposed algorithms, and compare their numerical effectiveness and performance on a variety of large-scale problems.

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Correspondence to Johannes J. Brust.

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This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344. R. Marcia’s research is partially supported by NSF Grant IIS 1741490. C. Petra also acknowledges support from the LDRD Program of Lawrence Livermore National Laboratory under Projects 16-ERD-025 and 17-SI-005.

J. J. Brust was formerly at University of California Merced, Merced, CA.

Appendix A

Appendix A


Section 2: Background

\({\mathbf {s}}_{k-1}={\mathbf {x}}_{k} - {\mathbf {x}}_{k-1} \qquad \qquad \qquad \qquad \quad {\mathbf {S}}_k =\displaystyle [ {\mathbf {s}}_{k-l} \,\, \cdots \,\, {\mathbf {s}}_{k-1}]\)
\({\mathbf {y}}_{k-1}=\nabla f({\mathbf {x}}_{k}) - \nabla f({\mathbf {x}}_{k-1}) \qquad \quad {\mathbf {Y}}_k = \displaystyle \left[ {\mathbf {y}}_{k-l} \,\, \cdots \,\, {\mathbf {y}}_{k-1}\right] \)
\({\mathbf {S}}_k^T {\mathbf {Y}}_k={\mathbf {L}}_k + {\mathbf {T}}_k \qquad \qquad \qquad \qquad \quad {\mathbf {D}}_k=\text {diag}({\mathbf {S}}_k^T{\mathbf {Y}}_k)\)
\({\mathbf {B}}^{{(k)}}_0={\gamma _{k}} {\mathbf {I}}_n \qquad \qquad \qquad \qquad \qquad \qquad {\mathbf {H}}_k=\mathbf {B}^{-1}_k\)
\(\gamma _{k}={\mathbf {y}}_{k-1}^T {\mathbf {y}}_{k-1} / {\mathbf {y}}_{k-1}^T {\mathbf {s}}_{k-1} \qquad \qquad \,\, \delta _{k} = {1/\gamma _k}\)
\({\mathbf {B}}_k=\gamma _k {\mathbf {I}}_n + \widehat{\varvec{\Psi }}_k \widehat{\varvec{\Xi }}_k \widehat{\varvec{\Psi }}_k^T \qquad \qquad \qquad \widehat{\varvec{\Psi }}_k = [ {\mathbf {S}}_k \ \ {\mathbf {Y}}_k]\)
\({\mathbf {H}}_k=\delta _k {\mathbf {I}}_n + \widehat{\varvec{\Psi }}_k \widehat{{\mathbf {M}}}_k \widehat{\varvec{\Psi }}_k^T\)
\(\widehat{\varvec{\Xi }}_k = \displaystyle \gamma _k\left[ \begin{array}{cc} - {\mathbf {S}}_k^T {\mathbf {S}}_k &{} - {\mathbf {L}}_k \\ - {\mathbf {L}}_k^T &{} \ \ \gamma _k {\mathbf {D}}_k \end{array}\right] ^{-1}\)
\(\widehat{{\mathbf {M}}}_k = -(\gamma _k^{2} \widehat{\varvec{\Xi }}_k^{-1} + \gamma _k\widehat{\varvec{\Psi }}_k^T \widehat{\varvec{\Psi }}_k)^{-1}\)

Section 3: Trust-Region Subproblem Solution without an Inequality Constraint

\({\mathbf {K}}= \displaystyle \left[ \begin{array}{c c} {\mathbf {B}}_k &{} {\mathbf {A}}^T \\ {\mathbf {A}} &{} {\mathbf {0}} \end{array} \right] \)                   \( \begin{array}{l} \varvec{\Omega }_k = ( {\mathbf {A}} {\mathbf {B}}_k^{-1} {\mathbf {A}}^T )^{-1} \\ \varvec{\Psi }_k =[ {\mathbf {A}}^T \ \ \ \widehat{\varvec{\Psi }}_k ]\end{array}\) 
\(\mathbf {K}^{-1} = \displaystyle \left[ \begin{array}{c c}{\mathbf {B}}_k^{-1} \!- {\mathbf {B}}_k^{-1}{\mathbf {A}}^T \varvec{\Omega }_k {\mathbf {A}} {\mathbf {B}}_k^{-1} \ \ &{} {\mathbf {B}}_k^{-1}{\mathbf {A}}^T \varvec{\Omega }_k \\ ({\mathbf {B}}_k^{-1}{\mathbf {A}}^T \varvec{\Omega }_k)^T \ \ &{} -\varvec{\Omega }_k \\ \end{array} \right] \) 
\({\mathbf {V}}_k = {\mathbf {B}}_k^{-1} \!-\! {\mathbf {B}}_k^{-1}{\mathbf {A}}^T \varvec{\Omega }_k {\mathbf {A}} {\mathbf {B}}_k^{-1}\) 
\({\mathbf {V}}_k = \delta _k {\mathbf {I}}_n + \varvec{\Psi }_k {\mathbf {M}}_k \varvec{\Psi }_k^T\) 
\({\mathbf {W}}_k = {\mathbf {B}}_k^{-1}{\mathbf {A}}^T \varvec{\Omega }_k\) 
\({\mathbf {M}}_k = \displaystyle \left[ \begin{array}{c c} - \delta _k^2 \varvec{\Omega }_k &{} - \delta _k\varvec{\Omega }_k {\mathbf {C}}_k\\ - \delta _k {\mathbf {C}}_k^T \varvec{\Omega }_k &{} \ \widehat{{\mathbf {M}}}_k \!-\! {\mathbf {C}}_k^T\varvec{\Omega }_k{\mathbf {C}}_k \end{array} \right] \) 
\({\mathbf {C}}_k = {\mathbf {A}}\widehat{\varvec{\Psi }}_k\widehat{{\mathbf {M}}}_k \) 

Section 4: Trust-Region Subproblem Solution with an \(\ell _2\)-Norm Inequality Constraint

\({\mathbf {H}}_k(\sigma ) = ({\mathbf {B}}_k + \sigma {\mathbf {I}})^{-1} \qquad \qquad \qquad \qquad \quad \,\,\, {\mathbf {H}}_k = {\mathbf {H}}_k(0) \)
\(\varvec{\Phi }_k(\sigma ) = {\mathbf {I}}_n - {\mathbf {A}}^T\varvec{\Omega }_k(\sigma ) {\mathbf {A}}{\mathbf {H}}_k(\sigma ) \qquad \qquad \varvec{\Phi }_k = \varvec{\Phi }_k(0) \)
\({\mathbf {H}}_k(\sigma ) = \frac{1}{\gamma _k + \sigma }{\mathbf {I}}_n + \widehat{\varvec{\Psi }}_k\widehat{{\mathbf {M}}}_k(\sigma )\widehat{\varvec{\Psi }}_k^T\)
\(\varvec{\Omega }_k(\sigma ) = ({\mathbf {A}}{\mathbf {H}}_k(\sigma ){\mathbf {A}}^T)^{-1}\)
\(\widehat{{\mathbf {M}}}_k(\sigma ) = -\big ((\gamma _k + \sigma )^2 \widehat{\varvec{\Xi }}_k^{-1} + (\gamma _k + \sigma )\widehat{\varvec{\Psi }}_k^T\widehat{\varvec{\Psi }}_k \big )^{-1}\)
\({\mathbf {V}}_k(\sigma ) = {\mathbf {H}}_k(\sigma ) - {\mathbf {H}}_k(\sigma ){\mathbf {A}}^T \varvec{\Omega }_k(\sigma ){\mathbf {A}} {\mathbf {H}}_k(\sigma )\)
\({\mathbf {V}}_k(\sigma ) = {\mathbf {H}}_k(\sigma ) \varvec{\Phi }_k(\sigma )\)
\({\mathbf {s}}(\sigma ) = - {\mathbf {H}}_k(\sigma ) \varvec{\Phi }_k(\sigma ) {\mathbf {g}}_k\)
\({\mathbf {s}}'(\sigma ) = - {\mathbf {H}}_k(\sigma ) \varvec{\Phi }_k(\sigma ) {\mathbf {s}}(\sigma )\)

Section 5: Trust-Region Subproblem Solution with a Shape-Changing Norm Inequality Constraint

\({\mathbf {U}}_k = -\varvec{\Psi }_k{\mathbf {M}}_k \varvec{\Psi }_k^T\)
\({\mathbf {A}}^T = \mathbf {Q}_{1} \mathbf {R}_{1}\qquad \qquad \qquad \qquad \qquad \qquad \,\, \mathbf {Q}_{1} \mathbf {Q}_{1}^T = {\mathbf {A}}^T ({\mathbf {A}} {\mathbf {A}}^T)^{-1} {\mathbf {A}} \)
\({\mathbf {P}} = {\mathbf {I}}_n - {\mathbf {A}}^T ({\mathbf {A}} {\mathbf {A}}^T)^{-1} {\mathbf {A}} \qquad \qquad \qquad {\mathbf {P}}\widehat{\varvec{\Psi }}_k = \widehat{{\mathbf {Q}}}_2\widehat{{\mathbf {R}}}_2 \)
\(\widehat{{\mathbf {V}}}_2\widehat{\varvec{\Lambda }}_k \widehat{{\mathbf {V}}}^T_2 = \widehat{{\mathbf {R}}}_2 (\widehat{{\mathbf {M}}}_k-{\mathbf {C}}_k^T\varvec{\Omega }_k{\mathbf {C}}_k) \widehat{{\mathbf {R}}}^T_2 \)
\(\mathbf {Q}_{2} = \widehat{{\mathbf {Q}}}_2 \widehat{{\mathbf {V}}}_2\)
\({\mathbf {Q}} = \left[ \mathbf {Q}_{1} \, \mathbf {Q}_{2} \, \mathbf {Q}_{3} \right] \)
\( \mathbf {Q}_{\parallel } = \left[ \mathbf {Q}_{1} \, \mathbf {Q}_{2} \right] \qquad \qquad \qquad \qquad \qquad \quad \mathbf {Q}_{\perp } = \mathbf {Q}_{3} \)
\({\mathbf {z}} = \left[ \begin{array}{c} \mathbf {z}_{1} \\ \mathbf {z}_{2} \\ \mathbf {z}_{3} \end{array} \right] \qquad \qquad \qquad \qquad \qquad \qquad \quad {\mathbf {s}} = {\mathbf {Q}} {\mathbf {z}}\)
\( \mathbf {z}_{\parallel } = \mathbf {z}_{2} = \mathbf {Q}_{2}^T {\mathbf {s}} \qquad \qquad \qquad \qquad \qquad \quad \mathbf {z}_{\perp } = \mathbf {z}_{3} = \mathbf {Q}_{3}^T {\mathbf {s}} \)
\( \mathbf {g}_{\parallel } = \mathbf {Q}_{2}^T {\mathbf {g}}_k \quad \qquad \qquad \qquad \qquad \qquad \qquad \mathbf {g}_{\perp } = \mathbf {Q}_{\perp }^T {\mathbf {g}}_k \)
\({\mathbf {V}}_k = {\mathbf {Q}} \varvec{\Lambda } {\mathbf {Q}}^T = \left[ \mathbf {Q}_{1} \, \mathbf {Q}_{2} \, \mathbf {Q}_{3} \right] \left[ \begin{array}{c c c} {\mathbf {0}} &{} \\ &{} \delta _k {\mathbf {I}} - \widehat{\varvec{\Lambda }}_k&{} \\ &{} &{} \delta _k {\mathbf {I}} \end{array} \right] \left[ \begin{array}{c} \mathbf {Q}_{1}^T \\ \mathbf {Q}_{2}^T \\ \mathbf {Q}_{3}^T \\ \end{array} \right] \)

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Brust, J.J., Marcia, R.F. & Petra, C.G. Large-scale quasi-Newton trust-region methods with low-dimensional linear equality constraints. Comput Optim Appl 74, 669–701 (2019).

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  • Linear equality constraints
  • Quasi-Newton
  • L-BFGS
  • Trust-region algorithm
  • Compact representation
  • Eigendecomposition
  • Shape-changing norm