Constrained Optimization

Constrained optimization optimizes an objective while requiring the solution to satisfy equality or inequality constraints. It appears in regularized training, normalization, simplex probabilities, KL-constrained post-training, molecular geometry, and constrained generation.

A general problem is:

$$\begin{array}{rl}\underset{x}{\min }& f(x)\\ \text{subject to}& h_i(x)=0,i=1,\dots ,m\\ & g_j(x)\le 0,j=1,\dots ,k\end{array}$$

The objective says what should be improved. The constraints say what must remain valid.

Equality Constraints and Lagrangian

For equality constraints $h(x)=0$, the Lagrangian is:

$$\mathcal{L}(x,\lambda )=f(x)+{\lambda }^{\top }h(x)$$

At a local optimum under regularity conditions:

$${\nabla }_x\mathcal{L}(x^{\text{*}},{\lambda }^{\text{*}})=0,h(x^{\text{*}})=0$$

The multiplier $\lambda$ measures how strongly the constraint affects the optimum.

Inequality Constraints and KKT Conditions

For inequality constraints $g_j(x)\le 0$, the Lagrangian is:

$$\mathcal{L}(x,\lambda ,\mu )=f(x)+{\lambda }^{\top }h(x)+{\mu }^{\top }g(x)$$

The Karush-Kuhn-Tucker conditions include:

$${\nabla }_x\mathcal{L}(x^{\text{*}},{\lambda }^{\text{*}},{\mu }^{\text{*}})=0$$ $$h(x^{\text{*}})=0,g(x^{\text{*}})\le 0$$ $${\mu }^{\text{*}}\ge 0,{\mu }_j^{\text{*}}{g}_j(x^{\text{*}})=0$$

The last condition is complementary slackness: an inactive inequality has zero multiplier.

Penalty and Barrier Forms

Many ML and molecular workflows do not solve constrained problems exactly. They turn constraints into penalties:

$$\underset{x}{\min }f(x)+\rho \sum _i{h}_i(x{)}^2+\rho \sum _j\max (0,g_j(x){)}^2$$

or use barriers that become large near invalid regions:

$$\underset{x}{\min }f(x)-\tau \sum _j\log (-g_j(x))$$

Penalties are easier to optimize, but they do not guarantee exact feasibility unless the protocol enforces it.

Projection View

Projected gradient descent takes an unconstrained step and projects back to the feasible set $\mathcal{C}$:

$$x_{t+1}={\Pi }_{\mathcal{C}}\left(x_t-\eta \nabla f(x_t)\right)$$

where:

$${\Pi }_{\mathcal{C}}(z)=\arg \underset{x\in \mathcal{C}}{\min }\Vert x-z{\Vert }_2^2$$

This is useful for simplex constraints, norm balls, valid coordinate sets, and constrained decoding abstractions.

Common AI and Computational Biology Examples

Setting	Constraint	Typical handling
probability vector	${\sum }_i{p}_i=1,p_i\ge 0$	softmax or simplex projection
KL-regularized policy	$D_{\mathrm{KL}}({\pi }_{\theta }\Vert {\pi }_{\mathrm{ref}})\le ϵ$	penalty, clipping, or trust region
molecular coordinates	bond lengths, clashes, chirality, pocket geometry	force field, projection, filtering
graph generation	valence, connectivity, atom types	constrained decoder or validity filter
normalization	mean/variance or norm constraint	reparameterization or layer normalization
resource-aware training	memory, latency, token budget	constrained model selection

Constraint vs Regularization

Hard constraint:

$$x\in \mathcal{C}$$

Soft regularization:

$$\underset{x}{\min }f(x)+\lambda R(x)$$

A regularizer encourages behavior; a constraint requires behavior. Many papers blur this distinction, so check whether invalid outputs are impossible, penalized, filtered, or silently removed.

Paper Reading Checks

• Is the constraint hard, soft, architectural, decoding-time, or post-hoc?
• Is feasibility guaranteed or only encouraged by a penalty?
• Are invalid samples counted in the denominator?
• Are constraint coefficients tuned on validation data?
• Does the constraint exist during deployment, or only during evaluation?
• Is the reported improvement from the model, projection, repair, or filtering?

Related

• Optimization
• Regularization
• Policy gradient
• Guidance
• Molecular generation
• Energy minimization
• Coordinate update
• Structured prediction