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A Spectral Projected Gradient Method for Computational Protein Design problem

Yukai Zheng, Qingna Li

TL;DR

This work tackles the computational protein design problem (CPD), a combinatorial optimization modeled as a $QSAP$ and equivalent to its continuous relaxation. It proposes two spectral projected gradient (SPG) based strategies, SCSC and SCP, to solve the CPD relaxation efficiently without directly forming the Hessian: SCSC solves the relaxation with simplex projection, while SCP uses a quadratic penalty to enforce equality constraints. The results show that both SPG-based methods outperform the state-of-the-art AQPPG in speed, often solving instances within 1 second, with SCSC delivering higher-quality solutions. Overall, the paper demonstrates that SPG, leveraging the spectrum of the local Hessian, provides a scalable and effective framework for large-scale CPD problems.

Abstract

In this paper, we consider the computational protein design (CPD) problem, which is usually modeled as 0/1 programming and is extremely challenging due to its combinatorial properties. As a quadratic semi-assignment problem (QSAP), the CPD problem has been proved to be equivalent to its continuous relaxation problem (RQSAP), in terms of sharing the same optimal objective value. However, since the current algorithm for solving this RQSAP uses the projected Newton method, which requires direct computation of the Hessian matrix, its computational cost remains quite high. Precisely for this reason, we choose to employ the spectral projected gradient (SPG) method to solve the CPD problem, whose effectiveness relies on choosing the step lengths according to novel ideas that are related to the spectrum of the underlying local Hessian. Specifically, we apply the SPG method in two distinct ways: direct solving the relaxation problem and applying a penalty method. Numerical results on benchmark instances verify the superior performance of our approach over the current algorithms in both quality and efficiency.

A Spectral Projected Gradient Method for Computational Protein Design problem

TL;DR

This work tackles the computational protein design problem (CPD), a combinatorial optimization modeled as a and equivalent to its continuous relaxation. It proposes two spectral projected gradient (SPG) based strategies, SCSC and SCP, to solve the CPD relaxation efficiently without directly forming the Hessian: SCSC solves the relaxation with simplex projection, while SCP uses a quadratic penalty to enforce equality constraints. The results show that both SPG-based methods outperform the state-of-the-art AQPPG in speed, often solving instances within 1 second, with SCSC delivering higher-quality solutions. Overall, the paper demonstrates that SPG, leveraging the spectrum of the local Hessian, provides a scalable and effective framework for large-scale CPD problems.

Abstract

In this paper, we consider the computational protein design (CPD) problem, which is usually modeled as 0/1 programming and is extremely challenging due to its combinatorial properties. As a quadratic semi-assignment problem (QSAP), the CPD problem has been proved to be equivalent to its continuous relaxation problem (RQSAP), in terms of sharing the same optimal objective value. However, since the current algorithm for solving this RQSAP uses the projected Newton method, which requires direct computation of the Hessian matrix, its computational cost remains quite high. Precisely for this reason, we choose to employ the spectral projected gradient (SPG) method to solve the CPD problem, whose effectiveness relies on choosing the step lengths according to novel ideas that are related to the spectrum of the underlying local Hessian. Specifically, we apply the SPG method in two distinct ways: direct solving the relaxation problem and applying a penalty method. Numerical results on benchmark instances verify the superior performance of our approach over the current algorithms in both quality and efficiency.

Paper Structure

This paper contains 13 sections, 1 theorem, 10 equations, 2 figures, 6 algorithms.

Table of Contents

  1. Introduction
  2. The CPD Problem
  3. Problem Formulation
  4. An Equivalent Continuous Relaxation Problem
  5. The SPG Method and Two Approaches for (\ref{['relaxation']})
  6. The Framework of SPG Method
  7. Two Approaches Based on the SPG Method
  8. SCSC
  9. SCP
  10. Numerical Results
  11. Details of the experimental setup
  12. Comparison with the state-of-the-art algorithm
  13. Conclusion

Key Result

Theorem 2.1

CPD Let $\Vert x\Vert_0$ denote the number of nonzero elements in $x$. There exists an optimal solution $x^*$ to the relaxation problem (relaxation) such that $\Vert x^*\Vert_0=n.$ Moreover, $x^*$ is also an optimal solution to the original problem (assignment).

Figures (2)

  • Figure 1: The relationship between positions and rotamers in the CPD problem
  • Figure 2: CPU time and iterations for each method

Theorems & Definitions (2)

  • Theorem 2.1
  • Remark 1