Need to help solving least sparse linear with some known values

Question

I have the problem describe below

I need to find value of x1', x2', x3', x4', x5' that make

(x1-x1')^2+(x2-x2')^2+(x3-x3')^2+(x4-x4')^2+(x5-x5')^2 = mininum value

and

x1' + x2' + x3' + x4' + x5' = 1

x1 + x2 + x3 + x4 + x5 = 1

Note: we know value of a, b, c, d, e, x1, x2, x3, x4, x5

Would anyone help me in this case?

I have tried with google/or-tools library but can't add condition to find minimum value.

    MPSolver solver = createSolver(solverType);
    double infinity = MPSolver.infinity();

    MPVariable x1 = solver.makeNumVar(0.0, infinity, "x1");
    MPVariable x2 = solver.makeNumVar(0.0, infinity, "x2");
    MPVariable x3 = solver.makeNumVar(0.0, infinity, "x3");
    MPVariable x4 = solver.makeNumVar(0.0, infinity, "x4");
    MPVariable x5 = solver.makeNumVar(0.0, infinity, "x5");

    // 0.15 <= x1 <= 0.35
    MPConstraint c1 = solver.makeConstraint(-infinity, 0.35);
    c1.setCoefficient(x1, 1);   
    MPConstraint c2 = solver.makeConstraint(0.15, infinity);
    c2.setCoefficient(x1, 1);

    // 0.1 <= x2 <= 0.3
    MPConstraint c3 = solver.makeConstraint(-infinity, 0.3);
    c3.setCoefficient(x2, 1);   
    MPConstraint c4 = solver.makeConstraint(0.1, infinity);
    c4.setCoefficient(x2, 1);

    // 0.0 <= x3 <= 0.2
    MPConstraint c5 = solver.makeConstraint(-infinity, 0.2);
    c5.setCoefficient(x3, 1);   
    MPConstraint c6 = solver.makeConstraint(0.0, infinity);
    c6.setCoefficient(x3, 1);

    // 0.15 <= x4 <= 0.35
    MPConstraint c7 = solver.makeConstraint(-infinity, 0.35);
    c7.setCoefficient(x4, 1);   
    MPConstraint c8 = solver.makeConstraint(0.15, infinity);
    c8.setCoefficient(x4, 1);

    // 0.1 <= x5 <= 0.3
    MPConstraint c9 = solver.makeConstraint(-infinity, 0.3);
    c9.setCoefficient(x5, 1);   
    MPConstraint c10 = solver.makeConstraint(0.1, infinity);
    c10.setCoefficient(x5, 1);

    // x1 + x2 + x3 + x4 + x5 = 1
    MPConstraint c11 = solver.makeConstraint(-infinity, 1.0);
    c11.setCoefficient(x1, 1);
    c11.setCoefficient(x2, 1);
    c11.setCoefficient(x3, 1);
    c11.setCoefficient(x4, 1);
    c11.setCoefficient(x5, 1);

    MPConstraint c12 = solver.makeConstraint(1.0, infinity);
    c12.setCoefficient(x1, 1);
    c12.setCoefficient(x2, 1);
    c12.setCoefficient(x3, 1);
    c12.setCoefficient(x4, 1);
    c12.setCoefficient(x5, 1);

    MPObjective objective = solver.objective();
    objective.setCoefficient(x1, 1);
    objective.setCoefficient(x2, 1);
    objective.setCoefficient(x3, 1);
    objective.setCoefficient(x4, 1);
    objective.setCoefficient(x5, 1);
    objective.setMinimization();

Answer 1

This is a basic constrain optimization problem with convex objective function. https://en.wikipedia.org/wiki/Constrained_optimization There are many softwares that help you to do this. e.g.
http://cvxopt.org/documentation/index.html