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3351Weighted Quadratic with Cheap x and Expensive yMinimize 1x 2+3y 2 subject to x+y\ge 4. Find (x *,y *) and the KKT multiplier.数学中等derivation未尝试面试订阅3356Feasible Center Means Zero MultiplierConsider minimizing (x-1) 2+(y-1) 2 subject to x+y\ge 1. At the optimum, is the inequality active or inactive, and what does KKT imply about ?数学中等derivation未尝试面试订阅3358Offset Point Already FeasibleConsider minimizing (x-2) 2+(y--1) 2 subject to x+y\ge 0. At the optimum, is the inequality active or inactive, and what does KKT imply about ?数学中等derivation未尝试面试订阅3360High y Coordinate Makes the Constraint SlackConsider minimizing (x--1) 2+(y-3) 2 subject to x+y\ge 1. At the optimum, is the inequality active or inactive, and what does KKT imply about ?数学中等derivation未尝试面试订阅3361Why Convexity Makes KKT So PowerfulWhy do KKT conditions become sufficient, not just necessary, in many convex optimization problems?数学中等essay未尝试面试订阅3362Complementary Slackness as an Economic StatementExplain complementary slackness in plain language to a PM who thinks of constraints as scarce resources.数学中等essay未尝试面试订阅3363When KKT Is Not Enough by ItselfGive one reason why solving the KKT equations in a nonconvex problem may fail to identify the global optimum.数学中等essay未尝试面试订阅3364Shadow Price InterpretationWhy is the optimal multiplier often interpreted as the marginal value of relaxing or tightening a constraint?数学中等essay未尝试面试订阅3365Why Slater-Type Regularity MattersWhy do regularity conditions such as Slater's condition matter when applying KKT?数学中等essay未尝试面试订阅3368Midpoint Rule Exact on a LineUsing a single-panel midpoint rule, approximate \int 0 4 (1+x)\,dx.数学简单derivation未尝试面试订阅3369Trapezoid Rule on a Symmetric Convex FunctionUsing a single-panel trapezoid rule, approximate \int -1 1 x 2\,dx.数学简单derivation未尝试面试订阅3370Midpoint Rule Misses a Symmetric BowlUsing a single-panel midpoint rule, approximate \int -1 1 x 2\,dx.数学简单derivation未尝试面试订阅3371Simpson on a CubicUsing one Simpson panel, approximate \int 0 2 x 3\,dx.数学中等derivation未尝试面试订阅3376Halving h in the Trapezoid RuleA smooth integrand is approximated with the trapezoid rule. If the current error at step size h is about 0.08, what is the rough error after halving the step size?数学中等derivation未尝试面试订阅3378Halving h in Simpson's RuleA smooth integrand is approximated with the Simpson rule. If the current error at step size h is about 0.016, what is the rough error after halving the step size?数学中等derivation未尝试面试订阅3381Degree of Exactness of Two-Point GaussFor two-point Gauss-Legendre quadrature on [-1,1], what is the highest polynomial degree integrated exactly?数学中等derivation未尝试面试订阅3382Two-Point Gauss on x^2+1Using two-point Gauss-Legendre on [-1,1], approximate \int -1 1 (x 2+1)\,dx.数学中等derivation未尝试面试订阅3383Two-Point Gauss on x^4Using two-point Gauss-Legendre on [-1,1], what approximation do you get for \int -1 1 x 4\,dx?数学中等derivation未尝试面试订阅3386Why Trapezoid Overestimates a Convex FunctionWhy does the trapezoid rule typically overestimate the integral of a convex function on one panel?数学中等essay未尝试面试订阅3387Why Midpoint Often Beats Trapezoid on Smooth ProblemsWhy can the midpoint rule outperform the trapezoid rule on smooth integrands even though both are second-order?数学中等essay未尝试面试订阅