‫بسم هللا الرحمن الرحيم‬
Kingdom of Saudi Arabia
Ministry of Higher Education
Majmaah University
‫ السابع‬-:‫المستوى‬
College Of Sciences in Alzulfi
OPER431-Z
-:‫تاريخ اإلمتحان‬
‫الـمـمـلكـة الـعـربـيـة الـسـعـوديـة‬
‫وزارة الـتـعـلـيـم الـعـالـي‬
‫جـامـعـة الـمـجـمـعـة‬
‫كلية العلوم بالزلفي‬
-:‫رقم ورمز المادة‬
‫ تقنيات األمثلية‬-:-‫اسم المادة‬
-:‫الرقم الجامعي‬
-:‫اسم الطالب‬
Answer the Following Questions
Q2) a-Determine whether the function 𝑓(𝑥1 , 𝑥2 ) = −𝑥1 2 − 3𝑥2 2 + 4𝑥1 𝑥2 + 10𝑥1 − 10𝑥2 is convex,
concave or neither at the point (0,0)
b - Find the value of θ such that the following Hessian matrix is PSD
2
𝐻 = [1
1
2
3
θ
3
2]
2
Q3) a) Let 𝑓: 𝑅 𝑛 → 𝑅 𝑚 and 𝑓: 𝑅 𝑛 → 𝑅 𝑘 be differentiable and convex .Let ∅: 𝑅 𝑚+𝑘 → 𝑅 satisfy the
following :If 𝑎1 ≥ 𝑎2 and 𝑏1 ≥ 𝑏2 , ∅(𝑎1 , 𝑏1 ) ≥ ∅(𝑎2 , 𝑏2 ).Consider the function ℎ: 𝑅 𝑛 → 𝑅 defined by
ℎ(𝑥) = ∅(𝑓(𝑥), 𝑔(𝑥)).Show that if ∅ is quasiconvex , ℎ is quasiconvex
b- Define a quasiconvex of a function and determine whether the following function is quasiconvex or
neither 𝑓(𝑥, 𝑦) = 𝑥 3 + 𝑦 3
Q1)a- Let 𝑓: 𝑅 𝑛 → 𝑅 be strictly quasiconvex. Consider the problem to minimize 𝑓(𝑥)subject to 𝑥 ∈
𝑆,where 𝑆 is nonempty convex set in 𝑅 𝑛 .Prove that if 𝑥̅ is a local optimal solution , 𝑥̅ is also global
optimal solution.
b-Let 𝑔𝑖 : 𝑅 𝑛 → 𝑅 𝑓𝑜𝑟 𝑖 = 1,2, … . , 𝑚. and let 𝑆be a nonempty open set in 𝑅 𝑛 .Consider the problem
𝑀𝑖𝑛 𝑓(𝑥) 𝑆𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜 𝑔𝑖 (𝑥) ≤ 0𝑓𝑜𝑟 𝑖 = 1,2, … . . , 𝑚, 𝑥 ∈ 𝑆
Let 𝑥̅ be a feasible point, and let 𝐼 = {𝑖: 𝑔𝑖 (𝑥̅ ) = 0}.Furthermore ,suppose that 𝑓 𝑎𝑛𝑑 𝑔𝑖 𝑓𝑜𝑟 𝑖𝜖𝐼 are
differentiable at 𝑥̅ and that 𝑔𝑖 𝑓𝑜𝑟 𝑖 ∉ 𝐼 is continuous at 𝑥̅ .Prove that if 𝑥̅ is a local optimal solution ,then
𝐹0 ∩ 𝐺0 = ∅,where 𝐹0 = {𝑑: ∇𝑓(𝑥̅ )𝑡 𝑑 < 0}, 𝐺0 = {𝑑: ∇𝑔𝑖 (𝑥̅ )𝑡 𝑑 < 0, 𝑓𝑜𝑟 𝑒𝑎𝑐ℎ 𝑖𝜖𝐼},
c- Show that the point 𝑥̅ = (2,1) is a local solution of the following problem
𝑀𝑖𝑛 𝑓(𝑥1 , 𝑥2 ) = (𝑥1 − 3)2 + (𝑥2 − 2)2
𝑆𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜 𝑥1 2 + 𝑥2 2 ≤ 5,
𝑥1 + 𝑥2 ≤ 3
𝑥1 ≥ 0, 𝑥2 ≥ 0
‫بسم هللا الرحمن الرحيم‬
‫الـمـمـلكـة الـعـربـيـة الـسـعـوديـة‬
‫وزارة الـتـعـلـيـم الـعـالـي‬
‫جـامـعـة الـمـجـمـعـة‬
‫كلية العلوم بالزلفي‬
‫اسم المادة‪ -:-‬تقنيات األمثلية‬
‫اسم الطالب‪-:‬‬
‫‪Kingdom of Saudi Arabia‬‬
‫‪Ministry of Higher Education‬‬
‫‪Majmaah University‬‬
‫رقم ورمز المادة‪-:‬‬
‫‪OPER431-Z‬‬
‫الرقم الجامعي‪-:‬‬
‫المستوى‪ -:‬السابع‬
‫‪College Of Sciences in Alzulfi‬‬
‫تاريخ اإلمتحان‪-:‬‬