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piecewise.m
function piecewise() % Copyright 2021, Gurobi Optimization, LLC % % This example considers the following separable, convex problem: % % minimize f(x) - y + g(z) % subject to x + 2 y + 3 z <= 4 % x + y >= 1 % x, y, z <= 1 % % where f(u) = exp(-u) and g(u) = 2 u^2 - 4 u, for all real u. It % formulates and solves a simpler LP model by approximating f and % g with piecewise-linear functions. Then it transforms the model % into a MIP by negating the approximation for f, which corresponds % to a non-convex piecewise-linear function, and solves it again. names = {'x'; 'y'; 'z'}; model.A = sparse([1 2 3; 1 1 0]); model.obj = [0; -1; 0]; model.rhs = [4; 1]; model.sense = '<>'; model.vtype = 'C'; model.lb = [0; 0; 0]; model.ub = [1; 1; 1]; model.varnames = names; % Compute f and g on 101 points in [0,1] u = linspace(0.0, 1.0, 101); f = exp(-u); g = 2*u.^2 - 4*u; % Set piecewise-linear objective f(x) model.pwlobj(1).var = 1; model.pwlobj(1).x = u; model.pwlobj(1).y = f; % Set piecewise-linear objective g(z) model.pwlobj(2).var = 3; model.pwlobj(2).x = u; model.pwlobj(2).y = g; % Optimize model as LP result = gurobi(model); disp(result); for v=1:length(names) fprintf('%s %d\n', names{v}, result.x(v)); end fprintf('Obj: %e\n', result.objval); % Negate piecewise-linear objective function for x f = -f; model.pwlobj(1).y = f; gurobi_write(model, 'pwl.lp') % Optimize model as a MIP result = gurobi(model); disp(result); for v=1:length(names) fprintf('%s %d\n', names{v}, result.x(v)); end fprintf('Obj: %e\n', result.objval); end