Static Optimisation

• Weierstrass Theorem: if S is a compact set, then a continuous f attains a global maximum and global minimum on S
• an interior optimum occurs at an interior point of S
• a constrained optimum occurs on the boundary of S, where the constraint binds
• if we have an unconstrained optimisation problem, or the set is open, we need only look for interior optima
• UO: if f has a local maximum or minimum at an interior point x* of S, then Df(x*)=0
• UO: 2nd order sufficient conditions - Hf(x*) neg def, x* = strict local max, Hf(x*) pos def, x* = strict local min, Hf(x*) indefinite = saddle point
• Eigenvalue Test: pos def if eigenvalues >0, neg def if eigenvalues < 0, weak inequalities for semi-def
• Sylvester's criterion: pd if all leading minors (det of submatrices from top left corner) are positive
• Sylvester's criterion: nd if -1^k . leading minors > 0 for all k
• in the case of concave functions, stationary points are automatically global maxima
• Hf is nd <=> f is strictly concave on S
• Hf is pd <=> f is strictly convex on S
• constraint qualification: the matrix Dh(x*) must be of full rank m where m is no. of constraints
• If the constraint qualification is satisfied we can look for a solution to our problem by solving n + m First Order Conditions for the Lagrangian
• the Lagrange multiplier measures the sensitivity of the value of the objective function at x* to a relaxation of the constraint h(x)=a
• If the constraint qualification fails we cannot apply the implicit function theorem, so Lagrangian approach might fail
• EC: if the last (n-m) leading principal minors of the bordered Hessian alternate, ending with -1^n, x* is a strict local maximum
• EC: if the last (n-m) leading principal minors of the bordered Hessian all have the same sign as -1^m, then x* is a strict local minimum
• Constraint qualification w/ inequalities: let gE be all the constraints which bind at x*, and e be the number of binding constraints. The constraint qualification its satisfied if the matrix DgE(x*) is of full rank e
• the Kuhn-Tucker theorem means that (provided the constraint qualification is satisfied) you can look for a solution to the optimisation problem by looking to a solution to (n+m) FOCs
• if a constraint doesn't bind, the corresponding Kuhn-Tucker Lagrange multiplier will be 0
• IC: if the sign of the last (n-e) leading principal minors is -1^e, constrained strict local minimum
• linear constraints never fail the constraint qualification
• IEC: if x* is a local maximum of f on S, and CQ holds, then x* is a solution to the FOCs, the equality constraints and the complementary slackness conditions for the inequality constraints
• if x0 is such that Df(x0)=0, then if f is concave, x0 is a global maximum
• if x0 is such that Df(x0)=0, then if f is convex, x0 is a global minimum
• interior critical (stationary) points are automatically global optima
• if f is strictly concave in S: if a stationary point exists, it is a (unique) global maximum and if a local max exists, it is a (unique) global max
• if f is strictly convex in S: if a stationary point exists, it is a (unique) global minimum and if a local min exists, it is a (unique) global min
• If P is a concave problem, then the Kuhn-Tucker FOCs are sufficient for a global maximum
• If P is a convex problem, then the Kuhn-Tucker FOCs are sufficient for a global minimum
• P: max f(x) s.t g(x)<=b is a concave problem if f is concave and each gj is convex (and vice versa)
• any strictly monotonic function of one variable is both (strictly) quasi-concave and (strictly) quasi-convex
• if f is strictly quasi-concave, an existent local maximum is a unique global maximum
• for concave problems the FOC are not only necessary but also sufficient for global optima