General Optimality Systems for PDE-Constrained Optimization

Overview¶

The previous lectures developed a complete theory for linear-quadratic elliptic optimal control:

• reduced formulation through the control-to-state map;

• adjoint-based gradient representation;

• variational inequality for control constraints;

• discrete KKT system and its numerical solution.

This lecture places these results in an abstract framework that covers nonlinear state equations and general objective functionals.

The logical path of the lecture:

1. fix the functional analytic setting — spaces, duality pairings, and adjoint operators;

2. define Fréchet differentiability of maps between Banach spaces;

3. derive the differentiability of the control-to-state map via the Implicit Function Theorem;

4. compute the Fréchet derivative of the reduced functional via the chain rule;

5. introduce the adjoint equation and rewrite the gradient without nested PDE solves;

6. formulate the Lagrangian, derive the full KKT system, and state the second-order conditions;

7. define the normal cone precisely and recover the variational inequality and projection formula;

8. compare the reduced and all-at-once approaches in terms of structure and computational cost;

9. verify the framework on a semilinear elliptic example.

Functional Analytic Setting¶

Let $X$, $Y$, $Z$, $U$ be real Banach spaces. A Banach space is a complete normed vector space. A Hilbert space is a Banach space whose norm is induced by an inner product: $\|v\|^2 = (v,v)$. Hilbert spaces are reflexive, meaning $(Y')' \cong Y$ isometrically.

Dual spaces and duality pairings. The dual space $X'$ consists of all bounded linear functionals $\ell: X \to \mathbb{R}$. For $\ell \in X'$ and $x \in X$ we write the duality pairing

The norm of $X'$ is $\|\ell\|_{X'} := \sup_{\|x\|_X \le 1} |\langle \ell, x \rangle|$.

Spaces used in this lecture.

• $Y$: the state space. For elliptic problems: $Y = H_0^1(\Omega)$ or $Y = H^1(\Omega)$. In the abstract framework: a Hilbert space.

• $U$: the control space. Typically: $U = L^2(\Omega)$ or $U = L^2(\Gamma)$. In the abstract framework: a Hilbert space.

• $Z'$: the constraint residual space, i.e., the codomain of the state equation. For elliptic problems: $Z' = H^{-1}(\Omega)$, the dual of $H_0^1(\Omega)$. In the abstract framework: the dual of some Banach space $Z$.

• $p \in Z$: the adjoint state or Lagrange multiplier for the state equation.

Adjoint operators. Let $A: X \to Z'$ be a bounded linear operator. Its adjoint $A^*: Z \to X'$ is defined by

If $X$ and $Z$ are Hilbert spaces and we identify $Z \cong Z'$ via the Riesz isomorphism, this becomes $(Ax, z)_Z = (x, A^* z)_X$.

Riesz isomorphism. In a Hilbert space $X$, the Riesz map $\mathcal{R}_X: X \to X'$ is defined by

It is an isometric isomorphism. Its inverse $\mathcal{R}_X^{-1}: X' \to X$ identifies the gradient: given $\ell \in X'$, the unique element $g := \mathcal{R}_X^{-1} \ell \in X$ satisfies

and we write $g = \nabla f(u)$ when $\ell = f'(u)$.

Fréchet Differentiability¶

Let $X$ and $W$ be Banach spaces, and let $F: X \to W$. $F$ is Fréchet differentiable at $x \in X$ if there exists a bounded linear operator $F'(x): X \to W$ such that

The operator $F'(x)$ is called the Fréchet derivative of $F$ at $x$.

For real-valued functionals $F: X \to \mathbb{R}$, the Fréchet derivative is an element of $X'$:

Partial derivatives. For a map $F: X_1 \times X_2 \to W$, the partial Fréchet derivative with respect to $X_1$ at $(x_1, x_2)$ in direction $h_1 \in X_1$ is

For fixed $(x_1, x_2)$, the map $h_1 \mapsto F_{x_1}(x_1, x_2) h_1$ is a bounded linear operator from $X_1$ to $W$.

Chain rule. If $G: X \to W$ and $H: W \to V$ are Fréchet differentiable, then $(H \circ G): X \to V$ is Fréchet differentiable and

For our purposes: given $S: U \to Y$ and $J: Y \times U \to \mathbb{R}$, with $f(u) := J(S(u), u)$,

Abstract Problem¶

We now state the abstract optimal control problem and fix notation for all derivatives.

Objective functional.

is assumed Fréchet differentiable, with partial derivatives

Constraint mapping.

is assumed Fréchet differentiable, with partial derivatives

These are bounded linear operators for each fixed $(y, u)$.

Admissible controls.

is a nonempty, closed, and convex subset.

Optimal control problem. Find

such that

Standing assumptions.

1. For every $u \in U_{\mathrm{ad}}$, the equation $e(y, u) = 0$ in $Z'$ admits a unique solution $y \in Y$.

2. $J$ and $e$ are of class $C^1$ (all Fréchet derivatives exist and are continuous).

3. At every feasible pair $(y, u)$, the operator $e_y(y, u): Y \to Z'$ is an isomorphism.

Assumption 3 is the regularity (constraint qualification) condition. For the Poisson state equation, it holds because $-\Delta: H_0^1(\Omega) \to H^{-1}(\Omega)$ is a homeomorphism.

Control-to-State Map¶

Under Assumption 1, define

where $S(u)$ is the unique solution of $e(y, u) = 0$ in $Z'$. The map $S$ is the control-to-state operator (or solution operator) of the state equation.

Differentiability via the Implicit Function Theorem. The Implicit Function Theorem in Banach spaces states: if $e: Y \times U \to Z'$ is $C^1$ and $e_y(y_0, u_0): Y \to Z'$ is an isomorphism at some feasible point $(y_0, u_0)$, then locally around $u_0$ the map $u \mapsto S(u)$ is $C^1$ and

where $y = S(u)$.

Derivation. Differentiate the identity $e(S(u), u) = 0$ with respect to $u$ in direction $h$:

Solving for $S'(u)h$ using invertibility of $e_y$:

This formula shows that:

• the state variation $S'(u)h$ due to a control perturbation $h$ is obtained by one linearized state solve;

• the linearized state equation is $e_y(y,u)[v] = -e_u(y,u)[h]$, driven by the perturbation;

• the map $h \mapsto S'(u)h$ is a bounded linear operator from $U$ to $Y$.

Example (linear Poisson model). For $e(y, u) = -\Delta y - u$:

Hence $S'(u)h = (-\Delta)^{-1}h = S(h)$, consistent with the linearity of $S$ in that case.

Reduced Functional and Chain Rule¶

Define the reduced functional

The PDE-constrained problem becomes an optimization problem in $u$ alone:

Fréchet derivative of $f$. By the chain rule,

where $y = S(u)$. Substituting the formula for $S'(u)h$:

This expression is not yet computationally convenient: evaluating the first term for every direction $h$ requires one linearized state solve per direction tested. The adjoint equation removes this bottleneck by moving the operator to the other argument.

Adjoint Equation¶

Setup. Let $(y, u)$ be a feasible pair. We want to rewrite the directional derivative $f'(u)h$ so that $h$ appears only in one bounded linear functional, without intermediate state solves.

Definition of the adjoint state. Introduce $p \in Z$ as the solution of the adjoint equation

The adjoint state $p$ is uniquely determined because $e_y(y, u)^*: Z \to Y'$ is an isomorphism (dual of the isomorphism $e_y(y,u): Y \to Z'$).

Note: $p$ lives in $Z$, the pre-dual of $Z'$. For the Poisson model, $Z = Z' = H^{-1}(\Omega)$ and the Riesz identification sends $p$ to $H_0^1(\Omega)$.

Gradient formula via the adjoint. With $p$ as above, use the duality identity: for any invertible $A: Y \to Z'$,

Apply this with $A = e_y(y,u)$, $q = J_y(y, u)$, $w = e_u(y, u) h$, and $p = (e_y^*)^{-1} J_y$:

Hence

Gradient of the reduced functional. Identifying $f'(u) \in U'$ with its Riesz representative $\nabla f(u) \in U$:

Cost count. Regardless of how many directions $h$ must be tested:

• one state solve to get $y = S(u)$;

• one adjoint solve to get $p$;

• one evaluation of $J_u - e_u^* p$.

Example (Poisson model). $J_y = y - y_d \in L^2(\Omega)$, $e_y^* = -\Delta$, so the adjoint equation is $-\Delta p = y - y_d$ with $p = 0$ on $\partial\Omega$. $J_u = \alpha u \in L^2(\Omega)$, $e_u^* = -I$, so $\nabla f(u) = p + \alpha u$, exactly as derived in Lecture 4.

Lagrangian Formulation¶

The Lagrangian naturally encodes both the objective and the constraint.

Definition.

Here $p \in Z$ is the Lagrange multiplier for the equality constraint $e(y, u) = 0$. The duality pairing is well-defined because $e(y, u) \in Z'$ and $p \in Z$.

Partial derivatives of $\mathcal{L}$.

With respect to $y$ in direction $v \in Y$:

With respect to $u$ in direction $h \in U$:

With respect to $p$ in direction $q \in Z$:

Interpretation of the conditions $\mathcal{L}_p = 0$, $\mathcal{L}_y = 0$.

• $\mathcal{L}_p(y, u, p) = 0$ means $e(y, u) = 0$ in $Z'$: the state equation.

• $\mathcal{L}_y(y, u, p) = 0$ means $e_y(y, u)^* p = J_y(y, u)$ in $Y'$: the adjoint equation.

• $\mathcal{L}_u(y, u, p) = 0$ (unconstrained case) means $J_u - e_u^* p = 0$ in $U'$: optimality.

First-Order Optimality System¶

We now state the full KKT conditions.

Unconstrained case ($U_{\mathrm{ad}} = U$). At a local minimizer $\bar u$ with corresponding state $\bar y = S(\bar u)$, there exists a unique adjoint state $\bar p \in Z$ such that:

Constrained case ($U_{\mathrm{ad}} \subsetneq U$). At a local minimizer $\bar u \in U_{\mathrm{ad}}$, the optimality condition for $u$ becomes the variational inequality

which in terms of the Lagrangian is

The full constrained KKT system is therefore:

The three equations live, respectively, in $Z'$, $Y'$, and $U'$. The three unknowns are $\bar y \in Y$, $\bar p \in Z$, and $\bar u \in U_{\mathrm{ad}}$.

Normal Cone and Variational Inequality¶

Definition. Let $K \subset U$ be a nonempty closed convex set and $u \in K$. The normal cone to $K$ at $u$ is the closed convex cone

Geometrically: $\xi \in N_K(u)$ if and only if $u$ maximizes the linear functional $v \mapsto \langle \xi, v \rangle$ over $K$ (i.e., $\xi$ points outward from $K$ at $u$).

Equivalence with the variational inequality. The normal-cone inclusion $-f'(\bar u) \in N_{U_{\mathrm{ad}}}(\bar u)$ is equivalent to

Projection. When $U$ is a Hilbert space and the Riesz identification $U \cong U'$ is used, for any $w \in U$ the projection $\Pi_K(w)$ is the unique element of $K$ satisfying

The normal-cone inclusion reads: $w - u \in N_K(u)$ if and only if $u = \Pi_K(w)$.

Projection formula for the control. The KKT condition $0 \in \nabla f(\bar u) + N_{U_{\mathrm{ad}}}(\bar u)$ is equivalent to

This is both a characterization of optimality and a natural fixed-point iteration (projected gradient step).

Box constraints in $L^2(\Omega)$. For

with $u_{\min}, u_{\max} \in L^\infty(\Omega)$, the normal cone at $\bar u \in U_{\mathrm{ad}}$ is characterized pointwise: $\xi \in N_{U_{\mathrm{ad}}}(\bar u)$ if and only if for a.e. $x \in \Omega$,

For the linear-quadratic case, the KKT condition $0 \in \alpha\bar u + \bar p + N_{U_{\mathrm{ad}}}(\bar u)$ is equivalent to the pointwise projection

Semismooth Newton for Control Constraints¶

The projection characterization of the KKT condition suggests a nonsmooth root equation. Fix $\rho > 0$ and define

Then

Hence solving the constrained first-order condition is equivalent to solving the nonsmooth equation $F(u)=0$ in $U$.

Proof of the three equivalences for $F(\bar u)=0$¶

We prove the equivalence chain in detail.

Fix $K := U_{\mathrm{ad}}$, and define

1. First equivalence

This follows directly from the definition $F(u)=u-\Pi_K(u-\rho\nabla f(u))$.

1. Second equivalence By the projection-normal-cone characterization in a Hilbert space, for any $u \in K$ and $w \in U$:

Applying this with $u=\bar u$ and $w=\bar u-\rho\nabla f(\bar u)$ gives

Since $N_K(\bar u)$ is a cone, $\eta \in N_K(\bar u)$ implies $(1/\rho)\eta \in N_K(\bar u)$ for every $\rho>0$, hence

Combining 1 and 2 yields

Semismoothness and generalized derivatives¶

In finite dimensions, a locally Lipschitz map $G: \mathbb{R}^n \to \mathbb{R}^m$ is semismooth at $x$ if for every sequence $h_k \to 0$ and every choice $V_k \in \partial_C G(x+h_k)$ (Clarke generalized Jacobian),

It is strongly semismooth if the remainder is $O(\|h_k\|^2)$.

In Hilbert spaces, the same idea is used with generalized derivatives of nonsmooth operators (Clarke/Bouligand derivatives or Newton derivatives). The key property is that each linearization captures the first-order behavior with a small remainder, so a Newton correction can be defined.

For box constraints, the projection is piecewise affine pointwise:

Therefore $\Pi_{[a,b]}$ is globally Lipschitz and strongly semismooth in finite dimensions, and so is the induced superposition operator in $L^q(\Omega)$ spaces ($1<q<\infty$).

Semismooth Newton step¶

Given an iterate $u^k$, choose a generalized derivative

and compute $\delta u^k \in U$ from

then update

For constrained PDE-control, each Newton step requires coupled evaluation of state and adjoint variables because $\nabla f(u)$ depends on $(y(u),p(u))$. In practice, one linearizes the reduced mapping $u \mapsto F(u)$ and solves the resulting linear system with PDE blocks using the same solver technology as in the all-at-once KKT setting.

Primal-dual construction with an additional multiplier for $F(u)=0$¶

The semismooth Newton step can be obtained from a KKT system with an explicit multiplier for the equation constraint $F(u)=0$.

At iteration $k$, consider the proximal feasibility problem

Its Lagrangian is

where $\lambda \in U$ is an additional Lagrange multiplier.

The first-order system is

Replace $F'(u^k)$ with a generalized derivative $M_k\in\partial F(u^k)$ and linearize at $(u^k,\lambda^k)$:

If we initialize and keep $\lambda^k=0$, the first block row gives $\delta u^k = -M_k^*\delta\lambda^k$, and substituting in the second row yields the Schur-complement equation

with

This primal-dual step is equivalent to solving a regularized Newton equation for $F(u)=0$ and leads to the same local model as the direct semismooth Newton correction when $M_k$ is nonsingular.

Hence semismooth Newton may be viewed in two equivalent ways:

• reduced form: solve $M_k\delta u^k=-F(u^k)$;

• primal-dual form: solve a saddle-point system in $(\delta u^k,\delta\lambda^k)$ with the extra multiplier enforcing the linearized equality constraint.

The primal-dual perspective is useful for block preconditioning and for embedding the step in all-at-once PDE-constrained solvers.

Explicit form for box-constrained linear-quadratic control¶

For

we have

and the root equation is

Define the active and inactive sets at iteration $k$ from

Then the generalized derivative of the projection is multiplication by the characteristic function $\chi_{\mathcal I^k}$, and the Newton step is equivalent to:

• enforce control bounds directly on active sets,

  $$u^{k+1}=u_{\min} \text{ on } \mathcal A_-^k, \qquad u^{k+1}=u_{\max} \text{ on } \mathcal A_+^k;$$

  (70)

• solve the unconstrained optimality equation on $\mathcal I^k$.

This is exactly the primal-dual active-set (PDAS) method. Thus, for box constraints, PDAS can be interpreted as a semismooth Newton method applied to the projection equation.

Local convergence statement¶

Under standard assumptions:

1. $\nabla f$ is (locally) Lipschitz differentiable in a neighborhood of $\bar u$;

2. $F$ is semismooth at $\bar u$;

3. every generalized derivative $M \in \partial F(\bar u)$ is uniformly invertible, i.e., $\|M^{-1}\| \le c$.

Then semismooth Newton is locally superlinearly convergent:

If $F$ is strongly semismooth and the derivative approximation is exact, one obtains local quadratic convergence:

These rates explain why active-set/Newton-type methods are often much faster than projected gradient iterations once the active set is close to the final one.

Reduced vs All-at-Once¶

The first-order optimality conditions can be approached in two complementary ways.

Reduced approach. The control $\bar u \in U_{\mathrm{ad}}$ is the only optimization unknown. The state $\bar y = S(\bar u)$ and adjoint $\bar p$ are functions of $\bar u$. The problem reduces to

and algorithms iterate on $u$ only, solving state and adjoint equations at each step.

At each optimization iterate $u^k$:

1. state solve: find $y^k \in Y$ such that $e(y^k, u^k) = 0$;

2. adjoint solve: find $p^k \in Z$ such that $e_y(y^k, u^k)^* p^k = J_y(y^k, u^k)$;

3. gradient: $g^k = \mathcal{R}_U^{-1}(J_u(y^k, u^k) - e_u(y^k, u^k)^* p^k) \in U$;

4. control update: $u^{k+1} = \Pi_{U_{\mathrm{ad}}}(u^k - \tau_k g^k)$.

Advantages:

• the optimization variable lives only in $U$;

• off-the-shelf iterative solvers can be reused for state/adjoint;

• memory footprint is one control vector.

Disadvantages:

• state and adjoint must be solved to sufficient accuracy at each step;

• second-order information (Hessian of $f$) requires a second adjoint solve (second-order adjoint);

• slow convergence when inner solves are expensive.

All-at-once approach. The triple $(\bar y, \bar u, \bar p)$ is the unknown. One applies Newton’s method directly to the full KKT system. The Newton step $(\delta y, \delta u, \delta p)$ solves the block system

where $\mathcal{L}_{yy}$, $\mathcal{L}_{yu}$, $\mathcal{L}_{uu}$ are second partial derivatives of the Lagrangian and $N_{U_{\mathrm{ad}}}^{-1}$ is the linearization of the normal-cone inclusion (active-set method or semismooth Newton regularization).

For the linear-quadratic case, the block system is the saddle-point system seen in Lecture 7, which is linear and can be solved directly. For nonlinear problems, Newton linearization is needed and the block structure is the same, but with variable-dependent operators.

Advantages:

• quadratic convergence near the solution (Newton’s method);

• one global solve per Newton iteration rather than many inner iterations;

• second-order information built in.

Disadvantages:

• the system lives in $Y \times U \times Z$, which is much larger than $U$ alone;

• saddle-point preconditioning is needed for efficiency.

A Worked Example: Semilinear Elliptic Control¶

We verify that the abstract framework applies to a nonlinear state equation.

Model problem. Let $\Omega \subset \mathbb{R}^d$ be a bounded Lipschitz domain and $d: \mathbb{R} \to \mathbb{R}$ a smooth function with $d(0) = 0$ (the semilinearity). Consider