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1996Convex Execution Cost With a Capacity Barrier 1A schedule pays a quadratic cost but also faces a blow-up term as it nears a hard capacity cap. Show that f(q) = 1 q 2 + 2/(1-q) is strictly convex on q<1.数学简单derivation未尝试免费1997Strict Convexity of a Barrier-Regularized Cost 2The desk wants a direct curvature argument, not a vague appeal to 'it looks bowl-shaped'. Show that f(q) = 3 q 2 + 1/(1-q) is strictly convex on q<1.数学简单derivation未尝试免费1998Convex Portfolio Penalty With Aggregate Exposure 3Each sleeve has its own quadratic penalty, and the whole book also pays for aggregate balance-sheet usage. Prove that F(w 1,w 2,w 3) = 2w 1 2 + 3w 2 2 + 5w 3 2 + 1(w 1+w 2+w 3) 2 is convex.数学中等derivation未尝试免费1999When the Portfolio Penalty Is Strictly Convex 4For F(w)=sum i a i w i 2 + gamma(sum i w i) 2 with all a i>0 and gamma>=0, is F strictly convex?数学中等derivation未尝试免费2002Shifted Log-Sum-Exp Convexity 7One stress term slopes down and the other slopes up, but the smooth envelope remains convex. Show that g(x) = ln(exp(-1x) + exp(3x + 0)) is convex on R.数学中等derivation未尝试免费2003Execution Cost With Stronger Capacity Wall 8The blow-up term is steeper, but the same convexity logic should still go through. Show that f(q) = 2 q 2 + 4/(1-q) is strictly convex on q<1.数学中等derivation未尝试免费2004Perspective Penalty 9A schedule trades size x over time t and pays x 2/t plus a linear time charge. Show that P(x,t)=x 2/t + 2 t is convex on the domain t>0.数学中等derivation未尝试免费2005Convexity of a Two-Asset Quadratic With a Balance-Sheet Term 10Show that H(w 1,w 2)=2w 1 2+5w 2 2+3(w 1+w 2) 2 is convex.数学困难derivation未尝试免费2008Barrier Cost With Strong Quadratic Curvature 13Both the quadratic inventory term and the capacity wall are active sources of curvature. Show that f(q) = 4 q 2 + 3/(1-q) is strictly convex on q<1.数学中等derivation未尝试免费2009Log Barrier Plus Ridge Penalty 14The desk penalizes approaching a utilization cap and also adds a quadratic regularizer. Show that r(x) = -ln(1-2x) + 1x 2 is convex on x < 0.5.数学困难derivation未尝试免费2011Joint Convexity of Size-Time Execution Cost 16The cost couples size and trading time through a perspective form. Show that P(x,t)=x 2/t + 3 t is convex on the domain t>0.数学简单derivation未尝试免费2012Convexity of a Tighter Utilization Penalty 17The utilization cap is tighter, but the same barrier argument applies. Show that r(x) = -ln(1-3x) + 2x 2 is convex on x < 0.333333.数学中等derivation未尝试免费2013Sum of Convex Terms Stays Convex 18If c 1(q)=q 2+1/(1-q) and c 2(q)=2q 2+3/(1-q), why is c 1(q)+c 2(q) convex on q<1?数学中等derivation未尝试免费2014Why the Smooth-Max Proxy Is Convex 19Explain in one sentence why log(exp(a 1 T x)+...+exp(a k T x)) is convex.数学困难derivation未尝试免费2015Inventory Cost Near a Hard Capacity Limit 20The PM wants a formal convexity check before using the function in an optimizer. Show that f(q) = 5 q 2 + 2/(1-q) is strictly convex on q<1.数学困难derivation未尝试面试订阅2019Convexity of Logistic Loss 24Show that ell(z)=ln(1+e -z ) is convex on R.数学中等derivation未尝试免费2020Convex Margin Penalty With a Soft Capacity Wall 25The margin term grows smoothly but sharply as the leverage coordinate approaches its cap. Show that r(x) = -ln(1-1x) + 3x 2 is convex on x < 1.数学困难derivation未尝试免费2023Square-Root Impact Averages Down 3A square-root impact proxy makes dispersion beneficial relative to plugging in the average size directly. Let psi(v)=sqrt(1+v) on v>=0. If V is random, how do E[psi(V)] and psi(E[V]) compare?数学中等derivation未尝试免费2024Square-Root Impact Gap for a Two-Scenario Slice 4A child-order size V is 0 with probability 1/2 and 3 with probability 1/2. Compute E[sqrt(1+V)] and sqrt(1+E[V]).数学中等数值题未尝试免费2029Exact Utilization Penalty Gap With Unequal Weights 9A utilization surcharge uses phi(q)=q/(1-q) on 0<=q<1. Suppose Q equals 0 with probability 1/3 and 3/4 with probability 2/3. Compute E[phi(Q)] and phi(E[Q]).数学困难数值题未尝试面试订阅