suggestions on solving the non-convex problem

[Jack1]

Given two column vectors v1, v2 of size n=20, I am trying to solve the following 

% Define decision variable

x = sdpvar(n,1);

a = sdpvar(2,1);

% Define an objective (minimize)

w_comb = rand(n,2)*a;     % <--- the rand(n,2) matrix would be a matrix of the two col. vectors (v1,v2)

Objective = w'*x - lambda*norm(x - x_old, 2)^2; 

% Define constraints 

Constraints = [-x_r<= x <= x_r, ...

            sum(a) == 1, ...

            a >= 0]; 

        

% Set some options for YALMIP and solver

options = sdpsettings('solver','mosek','verbose',0); 

% Solve the problem

sol = optimize(Constraints,-Objective,options);

I believe that my objective is non-convex due to two variables (a,x). 

Can I use an ADMM type approach to solve this problem? Any other alternatives would be much appreciated. Ty. 

[Johan Löfberg]

unclear what you are talking about. You define w_comb and then you never use it.

The problem you have (almost...) defined is a simple convex QP (asuming lambda >=0)

[Jack1]

apologies -- w_comb goes into the objective function instead of w.

If you could kindly suggest some ways, then, would be much appreciated. Ty.  

[Johan Löfberg]

well then nonconvex QP (of course, in you case with a 2d with one equality, it is a trivial problem)

https://yalmip.github.io/tags/#nonconvex-quadratic-programming