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In this JCP paper, the authors simultaneously enforce the discrete maximum principles and element-wise mass balance for advection-diffusion equations through convex optimization. The least-squares finite element method is used, so the problem becomes a mixed formulation with a flux variable $\boldsymbol{q}\in\mathbb{R}^{n_qdofs}$ and a scalar concentration $\boldsymbol{c}\in\mathbb{R}^{n_cdofs}$. The constraint optimization problem in the discrete setting is as follows:

$$\begin{align} \min_{q,c} \ &\tfrac12\langle c, K_{cc}c\rangle + \langle c, K_{cq}q\rangle + \tfrac12\langle q, K_{qq}q\rangle + \langle c,r_c \rangle + \langle q,r_q\rangle\tag{A}\\ \text{subject to: } &A_c c+ A_q q = b_f\tag{B}\\ &c_{\text{min}}\preceq c\preceq c_{\text{max}} \tag{C} \end{align} $$

where (B) is the equality constraints for element-wise mass balance and (C) is the bounded constraints to ensure discrete maximum principles. It is known that simply solving unconstrained version of (A) satisfies neither (B) nor (C), and the authors claim that this constrained optimization problem can satisfy both properties using MATLAB's quadprog function (interior-point-convex algorithm).

  1. If only (B) is enforced, there will still be violations in maximum principles, but the solution will be locally conservative.

  2. If only (C) is enforced, there will still be errors in mass balance, but the solution will satisfy the maximum principles.

  3. If both (B) and (C) are enforced, it makes sense to me that maximum principles will be met, but will element-wise mass balance necessarily be met? Based on what I understand about quadratic programming problems, the constraint (B) is associated with a lagrange multiplier $\lambda$ and is added to the objective functional (A), so if I am applying bounds to the new objective functional (A)+$\lambda\cdot$(B), wouldn't I still have some violations in mass balance (B) because my resulting objective functional value will be non-zero?

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This is just a standard convex Quadratic Programming (QP) problem, which quadprog, or any number of other QP solvers, can solve.

There are a variety of algorithms for solving QPs, and I think you need not concern yourself at this point with the algorithms. It appears that you are operating under a misunderstanding of the role and impact of Lagrange multipliers in solving the problem. Upon solution of the QP by the solver, you should have a solution which minimizes the objective function subject to constraints (B) and (C) being satisfied. How the QP solver accomplishes that is "its business". Don't be distracted by Lagrange multipliers, but yes, there will be Lagrange multipliers associated with each constraint at the solution. At the solution, it, along with the Lagrange multipliers, will satisfy the Karush-Kuhn-Tucker conditions, which are necessary and sufficient for an optimum for a convex QP.

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