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Monomial barrier functions for the box-constrained convex optimization problems

Hatem Fayed

TL;DR

A novel barrier function is introduced to convert the box-constrained convex optimization problem to an unconstrained problem and it is shown that despite LBFGSB was the fastest method for many problems, it failed to converge to the optimal solution for some problems and took a very long time to terminate.

Abstract

In this article, a novel barrier function is introduced to convert the box-constrained convex optimization problem to an unconstrained problem. For each double-sided bounded variable, a single monomial function is added as a barrier function to the objective function. This function has the properties of being positive, approaching zero for the interior/boundary points and becomes very large for the exterior points as the penalty parameter approaches zero. The unconstrained problem can be solved efficiently using Newton's method with a backtracking line search. Experiments were conducted using the proposed method, the interior-point for the logarithmic barrier (IP), the trust-region reflective (TR) and the limited-memory Broyden, Fletcher, Goldfarb, and Shanno for bound constrained problems (LBFGSB) methods on the convex quadratic problems of the CUTEst collection. Although the proposed method was implemented in MATLAB, the results showed that it outperformed IP and TR for all problems. The results also showed that despite LBFGSB was the fastest method for many problems, it failed to converge to the optimal solution for some problems and took a very long time to terminate. On the other hand, the proposed method was the fastest method for such problems. Moreover, the proposed method has other advantages, such as: it is very simple and can be easily implemented and its performance is expected to be improved if it is implemented using a low-level language, such as C++ or FORTRAN on a GPU.

Monomial barrier functions for the box-constrained convex optimization problems

TL;DR

A novel barrier function is introduced to convert the box-constrained convex optimization problem to an unconstrained problem and it is shown that despite LBFGSB was the fastest method for many problems, it failed to converge to the optimal solution for some problems and took a very long time to terminate.

Abstract

In this article, a novel barrier function is introduced to convert the box-constrained convex optimization problem to an unconstrained problem. For each double-sided bounded variable, a single monomial function is added as a barrier function to the objective function. This function has the properties of being positive, approaching zero for the interior/boundary points and becomes very large for the exterior points as the penalty parameter approaches zero. The unconstrained problem can be solved efficiently using Newton's method with a backtracking line search. Experiments were conducted using the proposed method, the interior-point for the logarithmic barrier (IP), the trust-region reflective (TR) and the limited-memory Broyden, Fletcher, Goldfarb, and Shanno for bound constrained problems (LBFGSB) methods on the convex quadratic problems of the CUTEst collection. Although the proposed method was implemented in MATLAB, the results showed that it outperformed IP and TR for all problems. The results also showed that despite LBFGSB was the fastest method for many problems, it failed to converge to the optimal solution for some problems and took a very long time to terminate. On the other hand, the proposed method was the fastest method for such problems. Moreover, the proposed method has other advantages, such as: it is very simple and can be easily implemented and its performance is expected to be improved if it is implemented using a low-level language, such as C++ or FORTRAN on a GPU.
Paper Structure (5 sections, 2 theorems, 33 equations, 4 figures, 10 tables, 2 algorithms)

This paper contains 5 sections, 2 theorems, 33 equations, 4 figures, 10 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. The monomial barrier function
  3. General framework
  4. Experimental results
  5. Conclusions

Key Result

Theorem 1

Suppose that $\boldsymbol{z}^k$ is the minimizer of $P(\boldsymbol{z}^k;\mu_k)$ in (eq101) above, and that $\mu_k \to \infty$. Then every limit point $\boldsymbol{z}$ of the sequence ${\boldsymbol{z}^k}$ is a solution of the problem (eq100).

Figures (4)

  • Figure 1: Graphs of $x^\mu/\mu$ for different values of $\mu$. $\mu=4$ () , $\mu=16$ () and $\mu=64$ ().
  • Figure 2: Graphs of $f(x)$ () and $P(x;64)$ () for Example 1 for different values of $a$.
  • Figure 3: Graphs of $y=0.8-x$(), $y=x^3$ () and $y=x^7$ ().
  • Figure 4: Contours of $P(\boldsymbol{x};\mu)$ and $P_l(\boldsymbol{x};1/\mu)$ for Example 2 for different values of $\mu$.

Theorems & Definitions (4)

  • Theorem 1: Convergence
  • proof
  • Theorem 2: Number of outer iterations
  • proof