how to do piecewise linear approximation

[lily ye]

If cost of a generation is  :

 cost(t)=a*p(t)^2+b*p(t)

we can approximate it by a set of piecewise blocks,only if we know the cost of p_min&p_max.BUT  I'm confused  how to figure out the number of the blocks i need to replace the p(t),if there are 3 piecewise blocks totally ? can i use"implies"? 

i don't know how to code it..

[Johan Löfberg]

https://yalmip.github.io/command/interp1/
https://yalmip.github.io/command/sos2/
https://yalmip.github.io/interp1qp

[Johan Löfberg]

but why are you using a pwa approximation of a convex quadratic cost? As long as you simply have that as an additive term in the objective, you will have a convex QP objective

[lily ye]

 "have that as an additive term in the objective" .  "That" means approximation or Quadratic cost? I've just learned  Yalmip less than  a month

[lily ye]

Thanks for your reply!

I see,So in my objective,I could type:

x=1:3:10; % 3 blocks in total

y=ax^2+bx;

sdpvar Cost P %P is the power of one generation at time k;

Cost=interp1(x,y,'linear',P);

is that rihgt?

[Johan Löfberg]

if your objective is ...+x^2+..., you can just as well keep that quadratic term in the objective, as it is convex and solvers such as cplex/gurobi/mosek are MIQP solvers and thus have no problems with convex quadratic expressions

[Johan Löfberg]

If you absolutely want to use a linear approximation of the quadratic cost, yes. My guess is the cost is convex though (a>=0) hence there is no reason to at least try to solve it as a MIQP before approximating it as a MILP

[lily ye]

If my objective is ax^2+bx+c,there is no need to transfer it?

If my constraints not only contain quadratic cost,but there is also standard SOCP,must I use a linear approximation?

MIQP+SOCP what kind solver can I use? How about MILP+SOCP?

[Johan Löfberg]

if a>=0, that is a convex quadratic and thus most integer solvers can deal with that

The second question makes no sense as you talk about cost in the constraint.

Mosek/Gurobi/CPLEX can all solve mixed-integer SOCP, which covers linear/convex quadratic/socp-representable constraints and costs