Convex Optimization Problem A convex optimization problem is one of the form minimize fo(x) subject to fi(x)<bi,i=1,...,m, where the functions fo,...,fm R"-R are convex,i.e.,satisfy fi(ax+By)<afi(x)+Bfi(y) for all,y∈R"and all a,3∈R with a+B=1,a≥0，B≥0. CLGO System Engineering by J.J.Gao 11
+ - System Engineering by J. J. Gao 11 Convex Optimization Problem
Brief History of Convex Optimization theory (convex analysis):ca1900-1970 algorithms 1947:simplex algorithm for linear programming (Dantzig) 1960s:early interior-point methods(Fiacco McCormick,Dikin,...) 1970s:ellipsoid method and other subgradient methods 1980s:polynomial-time interior-point methods for linear programming (Karmarkar 1984) late 1980s-now:polynomial-time interior-point methods for nonlinear convex optimization (Nesterov Nemirovski 1994) applications before 1990:mostly in operations research;few in engineering since 1990:many new applications in engineering (control,signal processing,communications,circuit design,...)new problem classes (semidefinite and second-order cone programming,robust optimization) CLGO System Engineering by J.J.Gao 12
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Today's Agenda Convex Optimization(Single Variable) Mini-Tutorial on Convex function ■MOO Formulation MOO:The weighting Method MOO:E-Constraint Method MOO:Goal Programming CLGO System Engineering by J.J.Gao 13
+ - 13 Today’s Agenda System Engineering by J. J. Gao Convex Optimization(Single Variable) Mini-Tutorial on Convex function MOO Formulation MOO: The weighting Method MOO:𝜖-Constraint Method MOO: Goal Programming
Mini Tutorial:Convex set line segment between z1 and z2:all points x=0x1+(1-0)x2 with0≤0≤1 convex set:contains line segment between any two points in the set x1,x2∈C,0≤0≤1→ 0x1+(1-0)x2∈C examples (one convex,two nonconvex sets) CLGO System Engineering by J.J.Gao 14
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Mini Tutorial:Convex set Convex Set Hyperplanes and halfspaces Example hyperplane:set of the form aT=b}(a0) 工0 aTz=b halfspace:set of the form{x|aTx≤b}(a≠0) a 工0 aTx≥b aTx≤b ·a is the normal vector hyperplanes are affine and convex;halfspaces are convex CLGO System Engineering by J.J.Gao 15
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