CSIR-UGC NET Optimization Techniques for Statisticians: practice questions

CSIR-UGC NET - Mathematical Sciences (statistics option) · India · National Testing Agency for CSIR

Original practice questions written to the published syllabus · not past papers

6worked here, free

29written to this exam in this subject

10syllabus units

What this subject covers

The optimisation theory statisticians use: linear programming with simplex and duality, transportation and assignment, integer and goal programming for farm and crop planning, non-linear optimisation with Kuhn-Tucker conditions, and the numerical optimisers inside likelihood estimation. The Agribusiness track teaches operations research for managers; this goes to the theory.

In the exams: AICE-JRF Ag Stat Unit 2 'linear programming: formulation, graphical, simplex, duality, transportation and assignment' (heavy); UPSC ISS Statistics III operations research (heavy, reported); ASRB NET Ag Stat (present)

Linear programming: formulation and graphical solution
Simplex method, Big-M and two-phase methods
Duality and sensitivity analysis
Transportation and assignment problems
Integer programming: branch and bound
Goal programming for multi-objective farm planning
Non-linear programming: Lagrange multipliers and Kuhn-Tucker conditions
Quadratic programming and risk programming (MOTAD)
Dynamic programming
Numerical optimisation: gradient, Newton and quasi-Newton methods

Where the 29 questions fall

How our questions for CSIR-UGC NET spread across the units of Optimization Techniques for Statisticians. Difficulty: 0 easy, 11 medium, 18 hard.

Numerical optimisation: gradient, Newton and quasi-Newton methods	6
Non-linear programming: Lagrange multipliers and Kuhn-Tucker conditions	5
Quadratic programming and risk programming (MOTAD)	3
Dynamic programming	3
Linear programming: formulation and graphical solution	2
Simplex method, Big-M and two-phase methods	2
Duality and sensitivity analysis	2
Transportation and assignment problems	2
Integer programming: branch and bound	2
Goal programming for multi-objective farm planning	2

6 practice questions, worked

Choose an answer, then open the working. Each explanation says why the right answer is right and why the others are not.

1 · Simplex method, Big-M and two-phase methods · hard

Why do modern solvers also use interior-point methods for very large LPs?

They always give integer solutions
They avoid the need for constraints
They give exact shadow prices without duality
Polynomial worst-case complexity

Answer and working

D. Polynomial worst-case complexity

Simplex is fast in practice but has exponential worst cases; interior-point methods have polynomial complexity and suit very large sparse problems.

Further reading: Taha, Operations Research: An Introduction; Hazell and Norton, Mathematical Programming for Economic Analysis in Agriculture; Nocedal and Wright, Numerical Optimization

2 · Goal programming for multi-objective farm planning · hard

With 12 ha and a 20-day labour limit, wheat (40/ha, 2 days) and mustard (30/ha, 1 day), can an income goal of 450 be met?

No; the maximum is 440
No; the maximum is 400
Yes, with 4 ha wheat and 8 ha mustard
Yes, with 12 ha wheat

Answer and working

A. No; the maximum is 440

Maximising 40x + 30y with x + y ≤ 12 and 2x + y ≤ 20 gives (8, 4) and 440 < 450, so the goals conflict and weights decide the compromise.

Further reading: Taha, Operations Research: An Introduction; Hazell and Norton, Mathematical Programming for Economic Analysis in Agriculture; Nocedal and Wright, Numerical Optimization

3 · Non-linear programming: Lagrange multipliers and Kuhn-Tucker conditions · medium

Which is NOT one of the KKT conditions for a maximum with constraints gᵢ(x) ≤ bᵢ?

∇f = Σμᵢ∇gᵢ
μᵢ ≥ 0
μᵢ[gᵢ(x) − bᵢ] = 0
μᵢ < 0 for binding constraints

Answer and working

D. μᵢ < 0 for binding constraints

Dual feasibility requires μᵢ ≥ 0; the others are stationarity and complementary slackness (plus primal feasibility).

Further reading: Taha, Operations Research: An Introduction; Hazell and Norton, Mathematical Programming for Economic Analysis in Agriculture; Nocedal and Wright, Numerical Optimization

4 · Dynamic programming · hard

Which methods solve a Markov decision process?

Branch and bound only
The simplex and MODI methods
Value and policy iteration
Newton-Raphson only

Answer and working

C. Value and policy iteration

Both iterate the Bellman equation for expected values.

Further reading: Taha, Operations Research: An Introduction; Hazell and Norton, Mathematical Programming for Economic Analysis in Agriculture; Nocedal and Wright, Numerical Optimization

5 · Numerical optimisation: gradient, Newton and quasi-Newton methods · medium

A mixed model reports non-convergence. Which cause is most common?

A bug in every optimiser
Over-parameterisation
Too many observations
Normal residuals

Answer and working

B. Over-parameterisation

Unidentifiable or boundary parameters make the objective flat or constrained; simplifying or reparameterising usually helps.

Further reading: Taha, Operations Research: An Introduction; Hazell and Norton, Mathematical Programming for Economic Analysis in Agriculture; Nocedal and Wright, Numerical Optimization

6 · Numerical optimisation: gradient, Newton and quasi-Newton methods · hard

After an MLE optimiser converges, why check that the Hessian is negative definite?

To make the gradient zero
To speed up the next run
To confirm a true maximum
To transform the parameters

Answer and working

C. To confirm a true maximum

A stationary point may be a saddle or minimum; the inverse of −H gives the standard errors only at a proper maximum.

Further reading: Taha, Operations Research: An Introduction; Hazell and Norton, Mathematical Programming for Economic Analysis in Agriculture; Nocedal and Wright, Numerical Optimization

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