关于fmincon在凸优化问题中初始点影响收敛结果的疑问

Great question—this is a common point of confusion when working with convex optimization in practice, especially when tools like fmincon don’t behave exactly as you’d expect from theory. Let’s break this down step by step:

First, the core rule: For strictly convex optimization problems, any reliable convex optimization algorithm (including the interior-point method used by default in fmincon) should converge to the unique global minimum, regardless of the initial point. For convex (but not strictly convex) problems, there may be multiple optimal points, but the objective function value at all these points will be identical.

So if you’re seeing different objective values (fval) from different initial points, one of three things is happening:

• Your problem isn’t actually convex (you might have made a mistake in the convexity analysis)
• Numerical precision issues are causing fmincon to stop at slightly different suboptimal points
• Non-smooth elements in your problem are throwing off the algorithm

Let’s look at your specific problem to diagnose:

a. Check Convexity of Your Objective & Constraints #

You stated your problem is convex, but let’s verify key parts:

• The transmission rate r_k = x_k * log2(1 + g_k*p_k/x_k) is a concave function of x_k (you can confirm this by computing the second derivative, which will be negative for x_k > 0).
• Your objective is sum( p_k*b_k / r_k )—since r_k is concave and positive, 1/r_k is a convex function (for positive concave functions that are increasing, their reciprocals are convex). So the objective should indeed be convex.
• Your nonlinear constraint has a max(in_s./r_ul) term. While the maximum of convex functions is convex, this introduces a non-smooth point (the gradient of the max function is discontinuous when two elements of in_s./r_ul are equal). fmincon’s interior-point method is designed for smooth convex problems; non-smoothness can cause the algorithm to converge to different points depending on the initial guess.

b. Numerical Precision Settings #

By default, fmincon uses relatively loose tolerance thresholds. If your optimal region is flat (or nearly flat), the algorithm might stop early at slightly different points that meet the default tolerance criteria but aren’t truly the global minimum.

Here are actionable steps to resolve your issue:

• Replace the non-smooth max constraint: Your current constraint max(in_s./r_ul) + in_e./r_ul - T1 ≤ 0 can be rewritten as a set of smooth constraints. The max term means the largest value of in_s./r_ul plus each in_e./r_ul must be ≤ T1. This is equivalent to for every user k, in_s./r_k + in_e./r_k ≤ T1 (since if the largest in_s./r_k satisfies this, all smaller ones will too). Removing the max eliminates non-smoothness, making the problem fully smooth and convex.
• Tighten fmincon’s numerical tolerances: Adjust the options to force the algorithm to converge more precisely:

  options = optimoptions('fmincon', ...
      'OptimalityTolerance', 1e-10, ...
      'ConstraintTolerance', 1e-10, ...
      'StepTolerance', 1e-10, ...
      'MaxFunctionEvaluations', 1e5, ...
      'MaxIterations', 1e4);
  problem.options = options;
  

• Use a dedicated convex optimization tool: If fmincon still gives inconsistent results, switch to a tool like CVX (a Matlab-based convex optimization modeling framework). CVX automatically verifies convexity, selects appropriate algorithms, and guarantees convergence to the global minimum for valid convex problems.

Here’s how to model your energy minimization problem in CVX:

user_num = size(p_ul, 1);
CVX_begin
    variable x(user_num) > 0  % Ensure x_k is positive to avoid division by zero
    minimize( sum( (in_s + in_e).*p_ul ./ (x .* log2(1 + a.*p_ul./x)) ) + sum(C1) )
    subject to
        sum(x) <= BW2;  % Total bandwidth constraint
        % Replace max constraint with per-user smooth constraints
        for k = 1:user_num
            (in_s + in_e(k)) ./ (x(k) * log2(1 + a(k)*p_ul(k)/x(k))) <= T1;
        end
CVX_end

% Access results:
b_ul = x;
fval = CVX_optval;

After making these changes, you should see consistent global optimal results regardless of the initial point. The key issue here was likely the non-smooth max constraint, which fmincon’s interior-point method struggles with. Switching to smooth constraints or using CVX will eliminate this problem.

内容的提问来源于stack exchange，提问作者user3919259