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Meijia Yang

Publications and source records attributed to Meijia Yang.

6 recordsLinked to original sources
arXiv2026

Global optimization of quadratic root-difference minimization under elliptic annulus constraints

This paper studies the nonconvex quadratic root-difference minimization under elliptic annulus constraints {\rm (QR)}. We first establish the Annulus Brickman theorem and equivalently reformulate {\rm (QR)} as a 2-dimensional convex problem {\rm (HP)} with hidden variables. We employ the Frank-Wolfe algorithm to globally solve {\rm (HP)}. A key finding is that the solutions of the Frank-Wolfe subproblems, which are traditionally viewed as mere auxiliary updates, are proven to be $O(1/\sqrt{k})$-approximate solutions of the original problem {\rm (QR)}. This transforms an algorithmic by-product into the primary output and completely bypasses the need to solve the computationally expensive quadratic system required for solution recovery. Leveraging this recovery-free property, we develop the efficient Iterative Minimum Generalized Eigenpair (IMGE) algorithm for globally solving {\rm (QR)}. Numerical experiments confirm that IMGE converges rapidly and significantly outperforms conventional methods, especially for large-scale problems.

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arXiv2024

Closing the duality gap of the generalized trace ratio problem

The generalized trace ratio problem {\rm (GTRP)} is to maximize a quadratic fractional objective function in trace formulation over the Stiefel manifold. In this paper, based on a newly developed matrix S-lemma, we show that {\rm (GTRP)}, if a redundant constraint is added and well scaled, has zero Lagrangian duality gap. However, this is not always true without the technique of scaling or adding the redundant constraint.

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arXiv2024

A polynomially solvable case of unconstrained (-1,1)-quadratic fractional optimization

In this paper, we consider an unconstrained (-1,1)-quadratic fractional optimization in the following form: $\min_{x\in\{-1,1\}^n}~(x^TAx+\alpha)/(x^TBx+\beta)$, where $A$ and $B$, given by their nonzero eigenvalues and associated eigenvectors, have ranks not exceeding fixed integers $r_a$ and $r_b$, respectively. We show that this problem can be solved in $O(n^{r_a+r_b+1}\log^2 n)$ by the accelerated Newton-Dinkelbach method when the matrices $A$ has nonpositive diagonal entries only, $B$ has nonnegative diagonal entries only. Furthermore, this problem can be solved in $O(n^{r_a+r_b+2}\log^2 n)$ when $A$ has $O(\log(n))$ positive diagonal entries, $B$ has $O(\log(n))$ negative diagonal entries.

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arXiv2019

Chebyshev Center of the Intersection of Balls: Complexity, Relaxation and Approximation

We study the n-dimensional problem of finding the smallest ball enclosing the intersection of p given balls, the so-called Chebyshev center problem (CCB). It is a minimax optimization problem and the inner maximization is a uniform quadratic optimization problem (UQ). When p<=n, (UQ) is known to enjoy a strong duality and consequently (CCB) is solved via a standard convex quadratic programming (SQP). In this paper, we first prove that (CCB) is NP-hard and the special case when n = 2 is strongly polynomially solved. With the help of a newly introduced linear programming relaxation (LP), the (SQP) relaxation is reobtained more directly and the first approximation bound for the solution obtained by (SQP) is established for the hard case p>n. Finally, also based on (LP), we show that (CCB) is polynomially solved when either n or p-n(> 0) is fixed.

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arXiv2018

A fast algorithm for globally solving Tikhonov regularized total least squares problem

The total least squares problem with the general Tikhonov regularization can be reformulated as a one-dimensional parametric minimization problem (PM), where each parameterized function evaluation corresponds to solving an n-dimensional trust region subproblem. Under a mild assumption, the parametric function is differentiable and then an efficient bisection method has been proposed for solving (PM) in literature. In the first part of this paper, we show that the bisection algorithm can be greatly improved by reducing the initially estimated interval covering the optimal parameter. It is observed that the bisection method cannot guarantee to find the globally optimal solution since the nonconvex (PM) could have a local non-global minimizer. The main contribution of this paper is to propose an efficient branch-and-bound algorithm for globally solving (PM), based on a novel underestimation of the parametric function over any given interval using only the information of the parametric function evaluations at the two endpoints. We can show that the new algorithm(BTD Algorithm) returns a global ε-approximation solution in a computational effort of at most O(n^3/ε) under the same assumption as in the bisection method. The numerical results demonstrate that our new global optimization algorithm performs even much faster than the improved version of the bisection heuristic algorithm.

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