niuyishuai / HDCA

Hybrid accelerated DCA for solving the Asymmetric Eigenvalue Complementarity Problem (AEiCP)

HDCA

Hybrid accelerated DCA for solving the Asymmetric Eigenvalue Complementarity Problem (AEiCP).

This project is supported by the National Natural Science Foundation of China (Grant No: 11601327).

Instruction

Run install.m to install the toolbox on MATLAB.

This toolbox is developed based on the DCAM toolbox, see here.

Three AEiCP models are considered. Seven DCA-type algorithms are implemented, including:

• BDCA with exact and inexact line search (BDCAe and BDCAa)
• ADCA
• InDCA
• Hybrid DCA with Line search and Inertial force (HDCA-LI)
• Hybrid DCA with Nesterov's extrapolation and Inertial force (HDCA-NI)
• The classical DCA

See the article here for more details about these models and algorithms.

Citation

@Misc{niu2023HDCA,
	title = {Accelerated DC Algorithms for the Asymmetric Eigenvalue Complementarity Problem},
	author = {Yi-Shuai Niu},	
	year = {2023},
	eprint={2305.12076},
        archivePrefix={arXiv},
	url = {https://arxiv.org/abs/2305.12076}
}

Dependencies

This toolbox depends on MOSEK for solving the linear and quadratic convex subproblems.

The compared optimization solvers are KNITRO, FILTERSD, IPOPT and MATLAB FMINCON. Please make sure that you have installed the corresponding solvers on MATLAB for comparison.

Samples

The Asymmetric Eigenvalue Complementarity Problem (AEiCP) involves finding complementary eigenvectors $x\in \mathbb{R}^n\setminus \{0\}$ and complementary eigenvalues $\lambda\in \mathbb{R}$ that satisfy the following conditions:

$$\begin{cases} w = \lambda B x - A x, \\ x^{\top} w = 0, \\ 0\neq x\geq 0, w\geq 0, \end{cases} $$

where $x^{\top}$ represents the transpose of $x$, $A\in \mathbb{R}^{n\times n}$ denotes an asymmetric real matrix, and $B$ is a positive definite matrix that doesn't necessarily have to be symmetric.

Here is a step-by-step example illustrating how to use different types of DCA algorithms to solve this problem:

• Generate a random AEiCP:

    n=10;
    randbnd=[-1,1];
    T=rand(n)*(randbnd(2)-randbnd(1))+randbnd(1);
    mu = -min([0,eigs(T+T',1,'smallestreal')]);
    A=T+mu*eye(n);
    B=eye(n);

• Set the initial point $x^0$:

We use random initialization

    x0.x = rand(n,1);
    x0.x = x0.x/sum(x0.x);
    x0.y = rand(n,1);
    x0.w = B*x0.x - A*x0.y;
    x0.z = sum(x0.y);

• Create a DC function object and a DC programming problem object:

    dcf=dcfunc;
    dcf.f=@(X,n,A,B,opt)fobj_eval_f1(X,n,A,B,opt);
    mydcp=dcp(dcf,[]);

In the function fobj_eval_f1, we use the first DC formulation (DCP1) proposed in the paper here, defined as:

$$0 = \min\{ f_1(x,y,w,z) := \Vert y-z x \Vert^2 + x^{\top}w : (x,y,w,z)\in \mathcal{C}_1 \},$$

where

$$\mathcal{C}_1: = \{(x,y,w,z)\in \mathbb{R}^{n} \times \mathbb{R}^{n} \times \mathbb{R}^{n} \times \mathbb{R}: w = Bx -A y, e^{\top}x = 1, e^{\top}y=z, (x,y,w,z)\geq 0 \},$$

and $f_1(x,y,w,z)$ has a DC-SOS decomposition $g_1(x,y,w,z) - h_1(x,y,w,z)$ where

$$ \begin{align} g_1(x,y,w,z) = &\Vert y\Vert^2 + \frac{((z+1)^2 + \Vert y-x\Vert^2)^2 + ((z-1)^2 + \Vert y+x\Vert^2)^2}{16} + \frac{(z^2 + \Vert x\Vert^2)^2}{2} + \frac{\Vert x+w\Vert^2}{4}, \\ h_1(x,y,w,z) = &\frac{((z+1)^2 + \Vert y+x\Vert^2)^2 + ((z-1)^2 + \Vert y-x\Vert^2)^2}{16} + \frac{z^4 + \Vert x\Vert^4}{2} + \frac{\Vert x-w\Vert^2}{4}. \end{align} $$

• Create and setup a DCA-type algorithm object

    mydca = dca(mydcp,x0);

    mydca.model='DCP1';	
    mydca.A=A;
    mydca.B=B;
    mydca.verbose=true;
    mydca.plot=false;
    mydca.tolf=0;
    mydca.tolx=0;
    mydca.maxiter=200;

    mydca.linesearch = false;
    mydca.nesterov = true;
    mydca.inertial = true;

mydca.maxiter is the maximum number of iterations for DCA-type algorithm, mydca.tolf and mydca.tolx are stopping tolerences such that we will terminate the algorithm if one of the stopping conditions $$\Vert X^{k+1}-X^k\Vert\leq \text{mydca.tolx} * (1+\Vert X^{k+1}\Vert) \quad \text{ or } \quad |f_1(X^{k+1})-f_1(X^k)|\leq \text{mydca.tolf} * (1+|f_1(X^{k+1})|)$$ is verified, where $X^k = (x^k,y^k,w^k,z^k)$. In this example, we set mydca.tolf=0, mydca.tolf=0 and mydca.xopt=200, which means the algorithm will terminate after 200 iterations.

Note that three boolean parameters mydca.linesearch, mydca.nesterov, and mydca.inertial are important for choosing different DCA-type algorithms. The settings in mydca for each DCA-type algorithm (DCA|ADCA|BDCAe|BDCAa|ADCA|InDCA|HDCA-LI|HDCA-NI) are summarized below:

Algorithms	Settings
DCA	linesearch = 0; nesterov = 0; inertial = 0;
BDCAe	linesearch = 1; nesterov = 0; inertial = 0; linesearch_type='exact';
BDCAa	linesearch = 1; nesterov = 0; inertial = 0; linesearch_type='armijo';
ADCA	linesearch = 0; nesterov = 1; inertial = 0; adca_q > 0; restartperiod = inf|>0;
InDCA	linesearch = 0; nesterov = 0; inertial = 1;
HDCA-LI	linesearch = 1; nesterov = 0; inertial = 1; linesearch_type='exact'|'armijo';
HDCA-NI	linesearch = 0; nesterov = 1; inertial = 1; adca_q > 0; restartperiod = inf|>0;

• Optimization

    status=mydca.optimize();

See more examples and advanced optional settings in the folder tests.

Available Dataset

Three datasets for AEiCP: RAND1, RAND2 and NEP are available.

See GEN_NEP.m and GEN_RANDEICP.m to generate these datasets.

License

Released under MIT license

Contact

[e-mail]