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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3388When Simpson's Rule Is Especially AttractiveWhen is Simpson's rule especially attractive compared with midpoint or trapezoid?数学中等essay未尝试面试订阅3389Why Oscillatory Integrals Need CareWhy can standard coarse-grid quadrature be unreliable on highly oscillatory integrals?数学中等essay未尝试面试订阅3390Why Adaptive Refinement Helps Near KinksWhy is adaptive refinement often better than a uniform fine grid when the integrand has a kink or localized sharp feature?数学中等essay未尝试面试订阅3391Central-Difference Richardson Upgrade 1A second-order central-difference estimate of f'(x) is 1.28 at h=0.2 and 1.22 at h=0.1. Using an O(h 2) Richardson extrapolation, what improved estimate do you get?数学简单derivation未尝试面试订阅