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gc_pwl.m
function gc_pwl % Copyright 2022, Gurobi Optimization, LLC % % This example formulates and solves the following simple model % with PWL constraints: % % maximize % sum c(j) * x(j) % subject to % sum A(i,j) * x(j) <= 0, for i = 1, ..., m % sum y(j) <= 3 % y(j) = pwl(x(j)), for j = 1, ..., n % x(j) free, y(j) >= 0, for j = 1, ..., n % % where pwl(x) = 0, if x = 0 % = 1+|x|, if x != 0 % % Note % 1. sum pwl(x(j)) <= b is to bound x vector and also to favor sparse x vector. % Here b = 3 means that at most two x(j) can be nonzero and if two, then % sum x(j) <= 1 % 2. pwl(x) jumps from 1 to 0 and from 0 to 1, if x moves from negative 0 to 0, % then to positive 0, so we need three points at x = 0. x has infinite bounds % on both sides, the piece defined with two points (-1, 2) and (0, 1) can % extend x to -infinite. Overall we can use five points (-1, 2), (0, 1), % (0, 0), (0, 1) and (1, 2) to define y = pwl(x) n = 5; % A x <= 0 A1 = [ 0, 0, 0, 1, -1; 0, 0, 1, 1, -1; 1, 1, 0, 0, -1; 1, 0, 1, 0, -1; 1, 0, 0, 1, -1; ]; % sum y(j) <= 3 A2 = [1, 1, 1, 1, 1]; % Constraint matrix altogether model.A = sparse(blkdiag(A1, A2)); % Right-hand-side coefficient vector model.rhs = [zeros(n,1); 3]; % Objective function (x coefficients arbitrarily chosen) model.obj = [0.5, 0.8, 0.5, 0.1, -1, zeros(1, n)]; % It's a maximization model model.modelsense = 'max'; % Lower bounds for x and y model.lb = [-inf*ones(n,1); zeros(n,1)]; % PWL constraints for k = 1:n model.genconpwl(k).xvar = k; model.genconpwl(k).yvar = n + k; model.genconpwl(k).xpts = [-1, 0, 0, 0, 1]; model.genconpwl(k).ypts = [2, 1, 0, 1, 2]; end result = gurobi(model); for k = 1:n fprintf('x(%d) = %g\n', k, result.x(k)); end fprintf('Objective value: %g\n', result.objval); end