OPT:010117The form e^x a^2+e^y b^2+e^{-x-y}(a+b)^2 is positive; AM-GM min 3 uniquely at 0; H(0) has eigs 1 and 3Convex Optimization Hessian Quadratic FormConvex Optimization Amgm Unique MinPrivateLabels unavailablePrivate saving unavailableConvex sets, convex functions, and epigraph geometryConvex Optimization · Mixed reviewProgress not loaded
OPT:010118LSE on R^3 has gradient p and Hessian diag(p)-pp^T; at 0 the eigs are 0,1/3,1/3 and the plane min is log 3 uniquely at 0Convex Optimization Lse Softmax GradientConvex Optimization Simplex Hessian SpectrumPrivateLabels unavailablePrivate saving unavailableConvex sets, convex functions, and epigraph geometryConvex Optimization · Mixed reviewProgress not loaded
OPT:010219The subdifferential of max{x1,x2,0} at 0 is the triangle conv{e1,e2,0}; (0.4,0.3) is valid and (0.8,0.4) is notConvex Optimization Max Affine SubdifferentialConvex Optimization Support MembershipPrivateLabels unavailablePrivate saving unavailableSeparation, conjugates, and subgradientsConvex Optimization · Mixed reviewProgress not loaded
OPT:010220The projection of (3,2) onto [-1,1]^2 is (1,1), distance sqrt(5); n=(2,1) supports at 3 and the midplane 11/2 is strictConvex Optimization Box Euclidean ProjectionConvex Optimization Strict Separating HyperplanePrivateLabels unavailablePrivate saving unavailableSeparation, conjugates, and subgradientsConvex Optimization · Mixed reviewProgress not loaded
OPT:010321The prox of -log is (v+sqrt(v^2+4 tau))/2; at v=1, tau=2 one obtains 2 with value -log 2+1/4, and the derivative stays in (0,1)Convex Optimization Log Barrier ProximalConvex Optimization Prox Derivative BoundPrivateLabels unavailablePrivate saving unavailableOptimality conditions, projections, and proximal operatorsConvex Optimization · Mixed reviewProgress not loaded
OPT:010322The unique minimizer of |x-1|+2|x+1|+(x-3)^2/2 is x=1 with value 6Convex One D Subdifferential FermatConvex Strong Convexity Unique MinimizerPrivateLabels unavailablePrivate saving unavailableOptimality conditions, projections, and proximal operatorsConvex Optimization · Mixed reviewProgress not loaded
OPT:010423The equality QP attains 2/5 at x*=(4/5,1/5) with multiplier nu*=-4/5 and zero gapConvex Equality Lagrange StationarityConvex Dual Function Zero GapPrivateLabels unavailablePrivate saving unavailableLagrangian duality, constraint qualifications, and KKT theoryConvex Optimization · Mixed reviewProgress not loaded
OPT:010424The LP and its dual both attain 10 uniquely at (2,2) and (2,1,0)Convex LP Dual ConstructionConvex Complementary Slackness UniquenessPrivateLabels unavailablePrivate saving unavailableLagrangian duality, constraint qualifications, and KKT theoryConvex Optimization · Mixed reviewProgress not loaded
OPT:010425Three logs on x1+x2+x3=6 attain 3 log 2 uniquely at (2,2,2)Convex Log Barrier Equality DualConvex Amgm UniquenessPrivateLabels unavailablePrivate saving unavailableLagrangian duality, constraint qualifications, and KKT theoryConvex Optimization · Mixed reviewProgress not loaded
OPT:010526Gradient descent on (x1^2+9x2^2)/2 with step 1/5 first meets f<=0.01 at k=14Convex Exact Gradient IterationConvex Least K Function TolerancePrivateLabels unavailablePrivate saving unavailableFirst-order, Newton, and splitting methods with ratesConvex Optimization · Mixed reviewProgress not loaded
OPT:010527Newton on x-log x is x(2-x); the error obeys e+=-e^2 and the orbit from 1/2 is 3/4, 15/16, 255/256View access optionsOpt Newton Scalar MapOpt Quadratic Error RecurrencePrivateLabels unavailablePrivate saving unavailableFirst-order, Newton, and splitting methods with ratesConvex Optimization · Mixed reviewProgress not loaded
OPT:010528Coordinate descent on (1/2)(x^2+xy+y^2) from (0,1) has yk=4^{-k} and fk=(3/2)16^{-k}View access optionsOpt Coordinate Descent Exact CycleOpt Quadratic Hessian SpectrumPrivateLabels unavailablePrivate saving unavailableFirst-order, Newton, and splitting methods with ratesConvex Optimization · Mixed reviewProgress not loaded
OPT:010629The SOC projection of (1,3,4) is (3,9/5,12/5); the residual in -K is orthogonal of norm 2sqrt2View access optionsOpt Soc Euclidean ProjectionOpt Moreau Polar ResidualPrivateLabels unavailablePrivate saving unavailableConic, second-order cone, and semidefinite optimizationConvex Optimization · Mixed reviewProgress not loaded
OPT:010630The robust value aTx+||PTx|| at (1,2) is 1+2sqrt2; robust<=4 is an SOC with slack 3-2sqrt2View access optionsOpt Robust Linear Dual NormOpt Soc Nonnegative RhsPrivateLabels unavailablePrivate saving unavailableConic, second-order cone, and semidefinite optimizationConvex Optimization · Mixed reviewProgress not loaded
OPT:010631The min enclosing circle of (-1,0),(1,0),(0,2) has center (0,3/4), radius 5/4, and KKT weights 5/16,5/16,3/8View access optionsOpt Min Enclosing Circle KktOpt Weighted Variance CertificatePrivateLabels unavailablePrivate saving unavailableConic, second-order cone, and semidefinite optimizationConvex Optimization · Mixed reviewProgress not loaded