Calculus & Optimization

Gradients, Lagrangians, convexity — how robots decide.

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Why it matters in robotics

Almost every decision a robot makes — a trajectory, a grasp, a control input, a sensor-fusion estimate — is the solution to an optimization problem: an objective to minimize subject to constraints. There are principled ways to handle those constraints directly, to tell whether a problem can be solved reliably or gets trapped in a bad partial answer, and to follow the slope of a function step by step toward the best solution. Turning a planning or control idea into a solvable formulation, and knowing which methods will actually converge, is the daily work of robotics.

What to study

✓Gradients, Jacobians, and Hessians; gradient descent and its convergence behavior (step size, conditioning, local vs. global minima)
✓Constrained optimization: Lagrange multipliers, the Lagrangian, and KKT conditions for equality/inequality constraints
✓Convexity: convex sets and functions, why convex problems have a global optimum, and recognizing LP/QP/SOCP/SDP standard forms
✓Applying it in robotics: formulating trajectory optimization and convex MPC, and solving problems with tools like CVXPY

Where to study

Cheatsheet

Symbols used below

f(x)

objective (cost) function

\nabla f

gradient (vector of partial derivatives)

\nabla^2 f

(or

H

)Hessian matrix (second derivatives)

\alpha

step size / learning rate

g_i, h_j

inequality / equality constraints

\lambda_i, \nu_j

Lagrange multipliers (dual variables)

L

Lagrangian

Gradient descent

$x_{k+1} = x_k - \alpha \nabla f(x_k)$

Steps downhill along the steepest local descent direction. On an ill-conditioned (elongated) cost surface it zig-zags, since it ignores curvature — convergence needs $\alpha$ small enough to not overshoot.

Newton's method

$x_{k+1} = x_k - [\nabla^2 f(x_k)]^{-1} \nabla f(x_k)$

Rescales the step by the inverse Hessian, so it accounts for curvature and heads more directly at the minimum — far fewer iterations near a well-behaved optimum, but each step costs more (Hessian build + solve).

Lagrangian & KKT conditions

For $\min f(x)$ s.t. $g_i(x) \le 0$ , $h_j(x) = 0$ : $L(x,\lambda,\nu) = f(x) + \sum_i \lambda_i g_i(x) + \sum_j \nu_j h_j(x)$

At an optimum: $\nabla_x L = 0$ (stationarity), $g_i(x) \le 0$ , $h_j(x)=0$ (feasibility), $\lambda_i \ge 0$ (dual feasibility), and $\lambda_i g_i(x) = 0$ (complementary slackness — a constraint is either active or its multiplier is zero).

Convexity

A function is convex if the line segment between any two points on its graph lies on or above the graph — equivalently $\nabla^2 f \succeq 0$ everywhere.

For a convex problem (convex objective, convex feasible set) any local minimum is the global minimum — this is why casting a problem as LP/QP/SOCP/SDP is so valuable: solvers can certify the global optimum.

Saddle points stall first-order methods

Near a saddle the gradient nearly vanishes even though it isn't a minimum — gradient descent crawls through these flat regions. Second-order methods see the mixed curvature (positive in some directions, negative in others) and escape faster.

Gradient descent steps downhill along −∇f. On an elongated (ill-conditioned) cost surface it zig-zags; second-order methods (Newton / Gauss–Newton) rescale by curvature and head more directly to the minimum.

	Gradient descent	Newton's method
Uses	Gradient only	Gradient + Hessian
Cost per step	Cheap: one gradient eval	Expensive: build + solve/invert Hessian
Convergence near optimum	Linear, sensitive to conditioning	Quadratic — very fast
Behavior on ill-conditioned surfaces	Zig-zags	Rescales by curvature, heads straight in

Near a saddle the gradient nearly vanishes, so a first-order method crawls: it descends one direction while the flat/ascending directions stall progress, until the trajectory finally slips off along a descent direction.

Application focus

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