Optimisation — the problem statement

Optimisation is the act of finding the best decision subject to constraints. Almost every quantitative-finance problem reduces to one: building the maximum-Sharpe portfolio, hedging an option book, calibrating a yield-curve model, finding the cheapest bond to deliver into a futures contract, designing a transaction-cost-aware execution schedule.

The canonical form

minimize       f(x)
subject to     g_i(x) ≤ 0,    i = 1, ..., m
               h_j(x) = 0,    j = 1, ..., p
over           x ∈ R^n

• x — the decision variables. Portfolio weights, hedge ratios, model parameters.
• f — the objective. Risk, cost, mis-pricing.
• g_i — inequality constraints. Long-only (w ≥ 0), exposure caps, turnover.
• h_j — equality constraints. Budget (Σw = 1), dollar-neutral, beta-target.

Local vs global optima

x* is a local minimum if f(x*) ≤ f(x) for all x in a neighbourhood; a global minimum if f(x*) ≤ f(x) for all feasible x. Finding global minima of arbitrary functions is hard. Convex problems are the exception: every local minimum is a global minimum. This is why convexity matters.

Existence

Weierstrass: if the feasible set is non-empty, closed, and bounded (compact), and f is continuous, then a global minimiser exists. Real-world finance problems often have unbounded feasible sets (no leverage limit, no short-sale constraint) — those problems can fail to have an optimum unless f penalises extreme positions.

Feasibility, optimality, sensitivity

• Feasibility: does any x satisfy all constraints?
• Optimality: among feasible x, which minimises f?
• Sensitivity: how does the optimum move when constraints or parameters shift?

Smoothness and differentiability

First-order methods (gradient, Newton) require f differentiable. Subgradient and proximal methods extend to non-smooth f (e.g., L¹ norm regularisation). Constraint smoothness matters analogously — interior-point methods need smooth boundary functions; barrier methods need strict feasibility.

Continuous, integer, mixed-integer

Variables can be continuous (portfolio weights), integer (number of shares, on/off indicators), or both (mixed-integer). Continuous convex problems are tractable; mixed-integer problems can be exponentially hard. Avoid integer constraints when continuous relaxations are close enough — and use branch-and-bound only when truly necessary.

Roadmap of this course

• Modules 2-3: convex foundations and unconstrained methods.
• Modules 4-5: Lagrangian, duality, KKT — the theory of constrained optimisation.
• Modules 6-7: LP and QP — closed-form and computationally tractable convex families.
• Modules 8-9: conic, robust, and stochastic optimisation — the modern toolkit.
• Module 10: solvers and software — turning math into running code.