Optimality Conditions for Elliptic Optimal Control with Constraints

Overview¶

The previous lecture established the continuous framework for linear-quadratic elliptic control:

• the reduced problem

  $$\min_{u\in Q} f(u),$$

  (1)

• the operatorial all-at-once formulation on

  $$V\times Q\times V, \qquad V:=H_0^1(\Omega),\quad Q:=L^2(\Omega),$$

  (2)

• the adjoint-based gradient formula

  $$\nabla f(u)=p+\alpha u,$$

  (3)

• and the unconstrained KKT system in dual spaces.

This lecture extends the framework to the case of control constraints.

The goal is to derive:

1. the variational inequality for box-constrained controls;

2. the operatorial KKT system with a nonlinear control condition;

3. the pointwise projection formula;

4. the saddle-point interpretation in infinite dimensions.

Notation and Setting¶

The operators are

with

The state equation is

and the cost functional is

Hence the all-at-once problem is

From the previous lecture, in the unconstrained case, the first-order system is

Equivalently, in reduced form,

This is the only fact we need to import from the previous lecture.

Admissible Controls¶

We now impose box constraints on the control:

We assume

so that $U_{\mathrm{ad}}\subset Q$ is nonempty, closed, and convex.

The constrained reduced problem is therefore

The state equation is unchanged. Only the admissible set for the control changes.

Variational Inequality¶

For minimization of a differentiable functional over a closed convex set in a Hilbert space, the first-order condition is a variational inequality rather than the equation $\nabla f(\bar u)=0$.

Thus $\bar u\in U_{\mathrm{ad}}$ is optimal if and only if

Using the reduced gradient from the previous lecture, this becomes

This is the constrained analogue of stationarity. If the constraint is inactive, one recovers $\bar p+\alpha \bar u=0$. If a bound is active, only admissible perturbations are allowed, so equality is replaced by an inequality.

Operatorial KKT System¶

The cleanest way to express the constrained problem in the operatorial setting is through the normal cone to $U_{\mathrm{ad}}$. For a closed convex set $K\subset Q$, the normal cone at $u\in K$ is

Then the variational inequality is equivalent to the inclusion

Therefore the full KKT system becomes

Compared with the unconstrained system from the previous lecture, only the third equation changes: the linear equation in $Q'$ is replaced by a nonlinear inclusion in $Q'$.

This can also be written as a block saddle-point relation:

in the dual product space

So the constrained problem still has the same KKT block structure as in the previous lecture, but now one block is nonlinear because of the cone term.

Projection Formula¶

For box constraints, the normal-cone inclusion has a pointwise characterization. It is equivalent to the projection formula

where $\Pi_{[u_{\min},u_{\max}]}$ denotes the pointwise projection onto the interval $[u_{\min}(x),u_{\max}(x)]$.

Explicitly, for almost every $x\in\Omega$,

Thus the control law remains explicit, but it is no longer linear. The unconstrained formula $u=-\alpha^{-1}p$ is simply clipped onto the admissible interval.

Full Optimality System¶

Combining state equation, adjoint equation, and control characterization, we obtain

In the Poisson model of this course, this reads

The first two equations are linear and exactly the same kind of objects as in the previous lecture. The third one carries the entire effect of the control constraint.

Interpretation as Infinite-Dimensional KKT¶

The constrained system is the direct analogue of finite-dimensional KKT conditions:

• the unknowns are functions in $V\times Q\times V$;

• the equations live in the dual product space $V'\times Q'\times V'$;

• the state and adjoint equations are linear operator equations;

• the control condition is a complementarity relation encoded either as a normal-cone inclusion or as a projection formula.

So the conceptual picture is:

• previous lecture: linear saddle-point system;

• current lecture: saddle-point system with a nonlinear control block.

This is exactly the structure later exploited by active-set and semismooth Newton methods.

Projected Gradient Methods¶

The most direct numerical method works on the reduced functional over the convex set $U_{\mathrm{ad}}$. Starting from $u^0\in U_{\mathrm{ad}}$, one computes at each iteration:

1. state solve

   $$A y^k - B u^k = F;$$

   (27)

2. adjoint solve

   $$M y^k - A^* p^k = M y_d;$$

   (28)

3. reduced gradient

   $$g^k=\nabla f(u^k)=p^k+\alpha u^k;$$

   (29)

4. projected update

   $$u^{k+1} = \Pi_{U_{\mathrm{ad}}}\!\left(u^k-\tau_k g^k\right).$$

   (30)

This is the Hilbert-space analogue of projected gradient descent for bound-constrained finite-dimensional optimization.

Two remarks are important:

• the projection enforces admissibility at every iteration;

• each step still requires one state solve and one adjoint solve, exactly as in the unconstrained reduced method.

For the linear-quadratic case, if one chooses

then

so the update becomes

This explains why the projection formula is both an optimality condition and a natural fixed-point iteration.

Projected gradient is robust and easy to implement, but it is usually only linearly convergent and may become slow when the active set is nearly identified but not yet fixed.

Primal-Dual Active Set Methods¶

The projection formula suggests splitting the domain into active and inactive regions. Given an iterate $(u^k,p^k)$, define

and the inactive set

To make the primal-dual structure explicit, introduce a multiplier

for the box constraints, and write the control optimality condition as

For box constraints, $\lambda$ satisfies the complementarity pattern

Once the active sets are frozen, the inequalities are replaced by equality constraints. On a given active-set guess $(\mathcal A_-^k,\mathcal A_+^k,\mathcal I^k)$, one may therefore consider the Lagrangian

where the equality constraints are imposed only on the active sets. The multipliers $\lambda_-^k$ and $\lambda_+^k$ are supported on $\mathcal A_-^k$ and $\mathcal A_+^k$, respectively.

Then the optimality conditions suggest the update rules

and

Equivalently, the multiplier can be recovered from

with

and sign restrictions on $\mathcal A_-^k$ and $\mathcal A_+^k$.

So, once the active sets are guessed, one solves a linear system for $(y^{k+1},u^{k+1},p^{k+1},\lambda^{k+1})$ with those sets frozen. In strong form this means:

Algorithmically:

1. guess active and inactive sets from the current iterate;

2. solve the equality-constrained KKT system with these sets fixed;

3. update the sets and repeat until they stop changing.

This is called a primal-dual active set method because:

• the primal variable $u$ determines where bounds are active;

• the dual variables $p$ and $\lambda$ decide whether a point should be active or inactive.

In practice, PDAS is often much faster than projected gradient because once the active set is identified, the remaining step is essentially the solution of the correct linearized KKT system.

The same idea extends beyond box constraints. Let $U_{\mathrm{ad}}\subset Q$ be any closed convex set for which the projection

is available in explicit or computational form. Then one may still write the optimality condition as

or equivalently as a normal-cone inclusion

and build active-set type methods by locally identifying the part of the constraint that behaves as an equality constraint at the current iterate. For box constraints this identification is pointwise and especially transparent, which is why PDAS takes its simplest form there. For more general admissible sets, the same principle survives, but the geometry of the active manifold and the structure of the projected Newton step become more problem-dependent.

Semismooth Newton Interpretation¶

The projection operator is not differentiable in the classical sense, but it is semismooth. This allows one to apply generalized Newton methods to the nonsmooth optimality system.

Recall the basic definition. Let $G:Q\to Q$ be locally Lipschitz and directionally differentiable. Its generalized derivative at $u$ is the Clarke generalized Jacobian

that is, the convex hull of all limits of classical derivatives computed at nearby points where $G$ is differentiable.

Then $G$ is called semismooth at $u$ if, for every perturbation $s\to 0$ and every generalized derivative $V_s\in \partial G(u+s)$ in the sense of Clarke,

If the stronger estimate

holds, one speaks of strongly semismoothness. Semismoothness is the regularity that replaces classical differentiability in generalized Newton theory.

A convenient reduced equation is

with some parameter $\sigma>0$. At a solution, this is equivalent to the variational inequality.

Indeed, in a Hilbert space the projection onto a closed convex set is characterized by

Apply this with

Then

Using the projection characterization, this is equivalent to

that is,

Since $\sigma>0$, this is equivalent to

which is exactly the variational inequality

So the fixed-point equation $G(u)=0$ and the constrained first-order condition are equivalent.

For box constraints, the projection acts pointwise, so its generalized derivative is easy to characterize:

• it is 0 on points that are projected onto an active bound;

• it is 1 on inactive points.

Therefore a generalized Newton step for $G(u)=0$ amounts to freezing the current active set and solving the corresponding linearized state-adjoint-control system. This is exactly the mechanism behind PDAS.

So, in the linear-quadratic box-constrained case, one can view:

• PDAS as an active-set method;

• semismooth Newton as a generalized Newton method for the projection equation;

• and, in this specific setting, the two viewpoints are essentially equivalent.

This equivalence is one of the main reasons why active-set methods are so effective for elliptic control problems with box constraints: they inherit Newton-type local behavior while preserving the simple geometric interpretation of active and inactive regions.

Algorithmic Comparison¶

The three methods fit the same state-adjoint-control structure, but they use it differently:

• projected gradient uses the projection only as a feasible descent step;

• PDAS uses the projection to identify active sets and then solves a linear KKT system;

• semismooth Newton interprets the projection equation as a nonsmooth root-finding problem.

Typical qualitative behavior:

• projected gradient: cheapest iteration, strongest robustness, slowest asymptotically;

• PDAS: more expensive iteration, often very fast once the active set is close to correct;

• semismooth Newton: Newton-type local convergence, but requires a good formulation and linear solver strategy.

This is why the continuous optimality system matters algorithmically: it tells us not only what the solution must satisfy, but also what class of numerical methods is natural for the problem.

Summary¶

This lecture builds directly on the previous one:

1. the state equation and adjoint equation are unchanged;

2. the unconstrained stationarity condition

   $$\alpha R_Q u+B^*p=0$$

   (60)

   is replaced by the constrained inclusion

   $$0\in \alpha R_Q u+B^*p+N_{U_{\mathrm{ad}}}(u);$$

   (61)

3. equivalently, the control satisfies the projection formula

   $$u=\Pi_{U_{\mathrm{ad}}}\!\left(-\frac1\alpha p\right);$$

   (62)

4. the full KKT system remains a saddle-point problem in dual spaces

   $$V'\times Q'\times V',$$

   (63)

   with a nonlinear control block.

5. this structure naturally leads to three algorithmic families:

   • projected gradient methods;

   • primal-dual active set methods;

   • semismooth Newton methods.

This closes the continuous first-order theory for linear-quadratic elliptic control with box constraints.

References¶

• Fredi Troeltzsch, Optimal Control of Partial Differential Equations, Chapter 2.

• Alfio Borzi and Volker Schulz, Computational Optimization of Systems Governed by Partial Differential Equations.

• Juan Carlos De Los Reyes, Numerical PDE-Constrained Optimization.