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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2010Smoothed Worst of Two Affine Stress Terms 15The worst-case proxy is no longer a hard max, but a smooth convex substitute. Show that g(x) = ln(exp(2x) + exp(-1x + 3)) is convex on R.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2016Why Convexity Matters Operationally 21Why do trading desks care whether an execution-cost model is convex rather than merely smooth?数学简单essay未尝试免费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2021Jensen Direction for a Utilization Penalty 1Randomizing a schedule changes the expected nonlinear penalty, not just the mean utilization. Let phi(q)=q/(1-q) on 0<=q<1. If Q is random with E[Q]=m, is E[phi(Q)] at least phi(E[Q]) or at most phi(E[Q])?数学简单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2025Reciprocal Funding Buffer Is Convex 5Let phi(L)=1/(1+L) on L>-1. State the Jensen inequality relation between E[phi(L)] and phi(E[L]).数学困难derivation未尝试面试订阅2026Why Dispersion Raises a Convex Penalty 6Why does randomizing Q around a fixed mean raise E[Q/(1-Q)] relative to plugging in the mean directly?数学简单derivation未尝试免费2027Log Barrier Jensen Direction 7A utilization score explodes as it nears 1, so the convexity direction matters for stress design. Let u(x)=-ln(1-x) on x<1. If U is random and almost surely below 1, compare E[u(U)] and u(E[U]).数学中等derivation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2031Backing Out the Stress State From a Reciprocal Plug-In Score 11A funding-buffer score uses phi(L)=1/(1+L). Suppose leverage L equals 0 with probability 1/2 and H with probability 1/2. If phi(E[L]) = 1/3, what is H and what is E[phi(L)]?数学简单derivation未尝试免费2032Barrier Score Gap for a Two-State Utilization Model 12Let u(x)=-ln(1-x) on x<1. Suppose U equals 0 with probability 1/2 and 3/4 with probability 1/2. Compute E[u(U)] and u(E[U]).数学简单数值题未尝试免费2034Conditional Jensen Lower Bound 14If phi is convex, what inequality holds between E[phi(X)|F] and phi(E[X|F]) almost surely?数学困难derivation未尝试面试订阅