How to warm start convex Quadratic Programs after changing objective function

Answered

Ruofeng Liu

January 11, 2025 20:43

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I am trying to use gurobipy to solve lasso-like problems. After solving the problem, I want to change one term in the linear part of the objective function and resolve using warm starts. I tried to do that by saving the VBasis, and CBasis, and then setting then after updating the model. However, Gurobi still does not use warm start in the resolve. I am wondering if am not doing things correctly, or is it just the case that warm start is not possible after changing the objective function?

def boxed_lasso_resolve(X, y, c, M, k):
    n, p = X.shape
    XtX = X.T @ X
    ytX = y.T @ X

    LASSO = gp.Model()
    beta = LASSO.addMVar(p, lb=-gp.GRB.INFINITY)
    x = LASSO.addVars(p)
    LASSO.setObjective(0.5 * beta.T @ XtX @ beta - ytX @ beta + 0.5 * np.dot(y, y) + gp.quicksum(c[i]*x[i] for i in range(p)), sense=gp.GRB.MINIMIZE)
    # constraints x[i]=|beta[i]|
    c1 = LASSO.addConstrs(-beta[j] <= x[j] for j in range(p))
    c2 = LASSO.addConstrs(beta[j] <= x[j] for j in range(p))
    # constraints -M<=beta[i]<=M
    c3 = LASSO.addConstrs(-beta[j] <= M for j in range(p))
    c4 = LASSO.addConstrs(beta[j] <= M for j in range(p))

    LASSO.Params.Method = 0 # 0=primal simplex, 1=dual simplex
    LASSO.optimize()

    # save VBasis
    VB_beta = beta.VBasis
    VB_x = [x[i].VBasis for i in range(p)]

    # save CBasis
    CB_c1 = [c1[i].CBasis for i in range(p)]
    CB_c2 = [c2[i].CBasis for i in range(p)]
    CB_c3 = [c3[i].CBasis for i in range(p)]
    CB_c4 = [c4[i].CBasis for i in range(p)]
    
    # modify objective
    c[k] = 0
    LASSO.setObjective(0.5 * beta.T @ XtX @ beta - ytX @ beta + 0.5 * np.dot(y, y) + gp.quicksum(c[i] * x[i] for i in range(p)), sense=gp.GRB.MINIMIZE)

    # load VBasis
    beta.VBasis = VB_beta
    for i in range(p):
        x[i].VBasis = VB_x[i]

    # load CBasis
    for i in range(p):
        c1[i].CBasis = CB_c1[i]
        c2[i].CBasis = CB_c2[i]
        c3[i].CBasis = CB_c3[i]
        c4[i].CBasis = CB_c4[i]

    LASSO.setParam("LPWarmStart", 1)
    LASSO.update()
    LASSO.optimize()

Related to

• Attribute
• Warm Start

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  Riley Clement

  • Gurobi Staff

  January 12, 2025 23:52

  Hi Ruofeng,

  I suspect the issue is the update() causing the CBasis and VBasis to be discarded. You can test this theory by inserting `print(LASSO.VBasis)` after the update() call.

  Can you reposition the update() after setting the new objective, but before setting the basis information and see if it resolves your issue?

  - Riley

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  Ruofeng Liu

  January 16, 2025 01:01

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  This solves my problem! Thank you very much!

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