Q1: Which optimization method is best suited for minimizing convex functions subject to inequality constraints?
(A) Interior-Point Method
(B) Genetic Algorithm
(C) Newton-Raphson Method
(D) Simplex Method
Answer: (A) Interior-Point Method
Q2: In a linear programming problem, the optimal solution lies:
(A) At the origin
(B) At the centroid of the feasible region
(C) At one of the vertices of the feasible region
(D) Anywhere in the feasible region
Answer: (C) At one of the vertices of the feasible region
Q3: The Kuhn-Tucker (KKT) conditions are used to solve:
(A) Linear optimization problems
(B) Unconstrained optimization problems
(C) Nonlinear constrained optimization problems
(D) Quadratic programming problems
Answer: (C) Nonlinear constrained optimization problems
Q4: Which of the following optimization methods uses random sampling and probability-based search?
(A) Gradient Descent
(B) Simplex Method
(C) Simulated Annealing
(D) Newton’s Method
Answer: (C) Simulated Annealing
Q5: The Wolfe conditions are used in:
(A) Line search methods for optimization
(B) Solving linear equations
(C) Nonlinear constraints
(D) Gradient-free optimization methods
Answer: (A) Line search methods for optimization
Applied Functional Analysis – MCQs with Answers
Q1: The Hahn-Banach theorem guarantees the existence of:
(A) A unique linear functional
(B) A bounded linear functional on a subspace
(C) A compact operator
(D) A dual space with finite dimension
Answer: (B) A bounded linear functional on a subspace
Q2: The spectrum of a bounded linear operator on a Hilbert space is:
(A) Always finite
(B) A subset of complex numbers
(C) Contained in the real line
(D) Equal to the eigenvalues of the operator
Answer: (B) A subset of complex numbers
Q3: A Banach space is defined as:
(A) A vector space with a bilinear form
(B) A complete normed vector space
(C) A normed space with an orthonormal basis
(D) A space of compact operators
Answer: (B) A complete normed vector space
Q4: In functional analysis, a compact operator is:
(A) Bounded and has a finite spectrum
(B) Linear and has a finite rank
(C) A mapping with a compact range
(D) Both bounded and maps bounded sets to relatively compact sets
Answer: (D) Both bounded and maps bounded sets to relatively compact sets
Q5: The Riesz Representation Theorem applies to:
(A) Banach spaces
(B) Hilbert spaces
(C) Sobolev spaces
(D) Compact spaces
Answer: (B) Hilbert spaces