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1969Which Side of Zero Is Optimal in the Linear-Softplus Problem 24For K(x)=m x - n ln(1+e x) with 0<m<n/2, is the unique optimizer positive, zero, or negative?数学中等derivation未尝试免费1970Maximum Value at the Interior Utility-Impact Optimum 25For F(x)=4 ln(1+x)-x 2 on x>-1, what x maximizes F and what is the maximum value?数学困难derivation未尝试免费1975Three-to-One Loading Tradeoff 5One book moves the target much faster per unit notional, so the optimizer must trade off load efficiency and cost. Minimize L(x,y) = 3x 2 + 1y 2 subject to 1x + 3y = 9.数学困难derivation未尝试面试订阅1978Why the Three-Book Solution Is Inverse-Coefficient Weighted 8In min a x 2 + b y 2 + c z 2 subject to x+y+z=N, why do larger quadratic coefficients receive smaller allocations?数学中等derivation未尝试免费1984Why the Total-and-Spread Problem Has a Unique Solution 14Why does the problem min a x 2 + b y 2 + c z 2 subject to x+y+z=N and x-z=d have a unique optimizer when a,b,c>0?数学困难derivation未尝试面试订阅1993Three-Book Allocation With Uneven Penalties 23The quadratic penalties differ across books, so the minimum-cost total-size allocation will be visibly asymmetric. Minimize Q(x,y,z) = 2x 2 + 3y 2 + 6z 2 subject to x+y+z = 22.数学简单数值题未尝试免费1995Alpha Maximization With Uneven Risk Units 25The second sleeve has both larger alpha and a different risk unit, so the optimal point must balance both effects. Maximize 4x + 6y subject to 4x 2 + 9y 2 = 225.数学困难derivation未尝试面试订阅2001Smooth Worst-Case Loss 6A desk smooths the max of two affine stress losses with a log-sum-exp surrogate. Show that g(x) = ln(exp(1x) + exp(2x + -1)) is convex on R.数学简单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2006Minimum Ridge for a Sharper Quartic Approximation 11A local approximation is even more non-convex, so the repair parameter must rise accordingly. A local PnL model is h(q)=q 4-10q 2+lambda q 2. What is the smallest lambda that makes h globally convex?数学简单数值题未尝试免费2007Three-Sleeve Convex Balance-Sheet Penalty 12The third sleeve is much more expensive on its own, but the aggregate term still preserves convexity. Prove that F(w 1,w 2,w 3) = 1w 1 2 + 4w 2 2 + 9w 3 2 + 2(w 1+w 2+w 3) 2 is convex.数学简单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2039Convex Penalty for Mixed Schedules 19Two execution schedules have penalties phi(q 1) and phi(q 2) under a convex phi. What does Jensen say about a random 50-50 mix versus the penalty at the average size?数学困难derivation未尝试面试订阅2046One Newton Step for a Log-Balance Equation 1A smooth calibration map mixes a linear term and a logarithmic correction. Apply one Newton step to solve x + 2 ln x = 3 starting from x 0 = 1.数学简单数值题未尝试免费2047Derive the Newton Update for x + a ln x = b 2Derive the Newton iteration for solving x + a ln x = b on x>0.数学简单derivation未尝试免费2048One Newton Step for a Reciprocal Balance Equation 3The desk solves a positive-state equation with both direct and reciprocal terms. Apply one Newton step to solve x + 2/x = 4 starting from x 0 = 2.数学中等数值题未尝试免费