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1966Balanced Softplus Tradeoff 21The linear reward and saturation penalty balance exactly at a central point. The desk maximizes K(x) = 2 x - 4 ln(1+e x). What x is optimal?数学简单数值题未尝试免费1967Positive Softplus Tilt 22The linear reward is strong relative to the saturation penalty, so the optimum should be positive. The desk maximizes K(x) = 3 x - 4 ln(1+e x). What x is optimal?数学简单derivation未尝试免费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1971Cheapest Two-Book Hedge 1Two hedge books carry different quadratic slippage penalties but must deliver one joint exposure target. Minimize L(x,y) = 1x 2 + 4y 2 subject to 1x + 2y = 10.数学简单数值题未尝试免费1972Derive the Two-Book Hedge Formula 2Derive the minimizer of a x 2 + b y 2 subject to u x + v y = c for positive a,b.数学简单derivation未尝试免费1976Three-Book Budget Allocation 6Three sleeves have different quadratic slippage penalties but must add up to a fixed total size. Minimize Q(x,y,z) = 1x 2 + 2y 2 + 4z 2 subject to x+y+z = 28.数学简单数值题未尝试免费1979Minimum Objective Value Under a Total-Size Constraint 9For positive a,b,c, what is the minimum value of a x 2 + b y 2 + c z 2 subject to x+y+z=N?数学中等derivation未尝试免费1981Three-Book Split With a Fixed Spread 11The first and third books must keep a pre-agreed spread while total size stays fixed. Minimize 1x 2 + 1y 2 + 1z 2 subject to x+y+z=9 and x-z=1.数学简单数值题未尝试免费1982Asymmetric Spread-Constrained Allocation 12The left book is more expensive per unit, so the optimizer cannot simply split the spread evenly. Minimize 2x 2 + 1y 2 + 1z 2 subject to x+y+z=10 and x-z=2.数学简单数值题未尝试免费1986Alpha Maximization on a Unit-Risk Ellipse 16Two sleeves carry different expected edges but must lie on one fixed quadratic risk budget. Maximize 3x + 4y subject to 1x 2 + 1y 2 = 25.数学简单数值题未尝试免费1987Derive the Alpha-Maximizing Point Under a Quadratic Risk Budget 17Derive the maximizer of mu 1 x + mu 2 y subject to a x 2 + b y 2 = R 2 for positive a,b.数学中等derivation未尝试免费1992Cheap-Fast Versus Expensive-Slow Hedge 22One hedge book is much cheaper, so the optimizer should lean on it heavily to hit the shared target. Minimize L(x,y) = 2x 2 + 8y 2 subject to 1x + 1y = 18.数学简单数值题未尝试免费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2000Ridge Needed to Repair Local PnL Curvature 5A research model has a locally non-convex quartic approximation, and risk wants the smallest ridge that fixes it everywhere. A local PnL model is h(q)=q 4-6q 2+lambda q 2. What is the smallest lambda that makes h globally convex?数学困难数值题未尝试免费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