Minimize[{x-y,-3 (x^2)+2 x y-(y^2)≥-1},{x,y}]

NMinimize[{x-y,-3 (x^2)+2 x y-(y^2)≥-1},{x,y}]

FindMinimum[{x-y,-3 (x^2)+2 x y-(y^2)≥-1},{x,y}]

FindMinimum[{- (100/((((x-1))^2)+(((y-1))^2)+1))-(200/((((x+1))^2)+(((y+2))^2)+1)),(x^2)+(y^2)>3 },{x,y}]

FindMinimum[{- (100/((((x-1))^2)+(((y-1))^2)+1))-(200/((((x+1))^2)+(((y+2))^2)+1)),(x^2)+(y^2)>3 },{{x,2},y}]

ContourPlot[- (100/((((x-1))^2)+(((y-1))^2)+1))-(200/((((x+1))^2)+(((y+2))^2)+1)),{x,-3,2},{y,-3,2},RegionFunction->(#1^2+#2^2>3&),Contours->10,Epilog->({Red,PointSize[.02],Text["global minimum",{-.995,-2.092}],Point[{-.995,-1.992}],Text["local minimum",{0.5304,1.2191}],Point[{1.2304,1.2191}]}),ContourLabels->True]

Show[{Plot3D[- (100/((((x-1))^2)+(((y-1))^2)+1))-(200/((((x+1))^2)+(((y+2))^2)+1)),{x,-4,3},{y,-4,3},RegionFunction->(#1^2+#2^2>3&),PlotRange->All],Graphics3D[{Red,PointSize[.02],Text["Global minimum",{-.995,-2.092,-230}],Point[{-.995,-2.092,-207}],Text["Local minimum",{0.5304,1.2191,-93.4}],Point[{1.23,1.22,-103.}]}]}]

Options of the (RefLink[FindMinimum, paclet:ref/FindMinimum]) function.

FindMinimum[{(x^2) + (y^2),(((x-1))^2)+2(((y-1))^2) > 5},{x,y},Method->"InteriorPoint"]

pts=Reap[sol=FindMinimum[{- (100/((((x-1))^2)+(((y-1))^2)+1))-(200/((((x+1))^2)+(((y+2))^2)+1)),(x^2)+(y^2)>3 },{{x,1.5},{y,-1}},MaxIterations->10, StepMonitor:> (Sow[{x,y}])];]; sol

ContourPlot[- (100/((((x-1))^2)+(((y-1))^2)+1))-(200/((((x+1))^2)+(((y+2))^2)+1)),{x,-3,3},{y,-4,2},RegionFunction->(#1^2+#2^2>3&),Contours->10,Epilog->({Red,PointSize[.01],Line[pts[[2,1]]],Yellow, Point/@ pts[[2,1]],Blue,PointSize[.02],Point[pts[[2,1,1]]]}),ContourLabels->True]

sol=FindMinimum[{-(100/((x-1)^2+(y-1)^2+1))-200/((x+1)^2+(y+2)^2+1),x^2+y^2>3},{{x,3/2},{y,-1}},WorkingPrecision->100,Method->"InteriorPoint"]

solExact=Minimize[{- (100/((((x-1))^2)+(((y-1))^2)+1))-(200/((((x+1))^2)+(((y+2))^2)+1)),(x^2)+(y^2)>3 },{x,y}];

sol[[1]]-solExact[[1]]

Plot[Sin[x]+.5 x,{x,-10,10}]

FindMinimum[{Sin[x]+.5 x,-10<=x<=10},{x}]

FindMinimum[{Sin[x]+.5 x,-10<=x<=10},{{x,-5}}]

FindMinimum[{Sin[x]+.5 x,-10<=x<=10&& x < -5},{x}]

SeedRandom[7919];
Table[FindMinimum[{Sin[x]+.5 x,-10<=x<=10},{{x,RandomReal[{-10,10}]}}],{10}]

NMinimize[{Sin[x]+.5 x,-10<=x<=10},{x},Method->{"RandomSearch","PostProcess"->"InteriorPoint"}]

(*FindMinMax[{Max[{f1,f2,..}],constraints},vars]*)
SetAttributes[FindMinMax, HoldAll];
FindMinMax[{f_Max, cons_}, vars_, opts___?OptionQ] := With[{res = iFindMinMax[{f, cons}, vars, opts]}, res /; ListQ[res]];
iFindMinMax[{ff_Max, cons_}, vars_, opts___?OptionQ] := Module[{z, res, f = List @@ ff},
   res = FindMinimum[{z, (And @@ cons) && (And @@ Thread[z >= f])}, Append[Flatten[{vars}, 1], z], opts]; If[ListQ[res], {z /. res[[2]], Thread[vars -> (vars /. res[[2]])]}]];

FindMinMax[{Max[{x^2,(x-1)^2}],{}},{x,y}]

FindMinMax[{Max[{Abs[2 x^2 + y^2 - 48 x - 40 y + 304], Abs[-x^2 - 3 y^2], Abs[x + 3 y - 18], Abs[-x - y],Abs[ x + y - 8]}], {}}, {x, y}]

ContourPlot[Max[{Abs[2 x^2 + y^2 - 48 x - 40 y + 304], Abs[-x^2 - 3 y^2], Abs[x + 3 y - 18], Abs[-x - y],Abs[ x + y - 8]}], {x, 3, 7}, {y, 1, 3}, Contours->40, Epilog->{Red, PointSize[0.02], Point[{4.93, 2.08}]}]

FindMinMax[{Max[{Abs[2 x^2 + y^2 - 48 x - 40 y + 304], Abs[-x^2 - 3 y^2], Abs[x + 3 y - 18], Abs[-x - y],Abs[ x + y - 8]}], {(x-6)^2+(y-1)^2≤1}}, {x, y}]

ContourPlot[Max[{Abs[2 x^2 + y^2 - 48 x - 40 y + 304], Abs[-x^2 - 3 y^2], Abs[x + 3 y - 18], Abs[-x - y],Abs[ x + y - 8]}], {x, 3, 7}, {y, 0, 3}, RegionFunction->(((#1-6)^2+(#2-1)^2<=1)&),Contours->40,Epilog->{Red,PointSize[0.02],Point[{5.34,1.75}]}]

(* GoalProgrammingWeightedAverage[{{{f1,goal1,weight1},...},cons},vars]]*)
GoalProgrammingWeightedAverage[{fg:{{_,_}..},cons_},vars_,opts___?OptionQ]:=With[
{res=Catch[iGoalProgrammingWeightedAverage[{Map[(Append@@#&),Thread[{fg,ConstantArray[1,{Length[fg]}]}]],cons},vars]]},
res/;ListQ[res]
];
GoalProgrammingWeightedAverage[{fg:{{_,_,_}..},cons_},vars_,opts___?OptionQ]:=With[
{res=Catch[iGoalProgrammingWeightedAverage[{fg,cons},vars]]},
res/;ListQ[res]
];
iGoalProgrammingWeightedAverage[{fg:{{_,_,_}..},cons_},vars_,opts___?OptionQ]:=Module[
{fs,goals,zs,z,res,ws},
{fs,goals,ws} = Transpose[fg];
If[!VectorQ[ws,(#>=0&)],Throw[$Failed]];
If[!VectorQ[goals,((NumericQ[#]&&Head[#]=!=Complex)&)],Throw[$Failed]];
zs=Array[z,Length[fs]];
res=FindMinimum[{ws.zs,(And@@Flatten[{cons},1])&&(And@@Thread[zs≥0])&&(And@@Thread[zs≥fs-goals ])},Join[Flatten[{vars},1],zs],opts];
If[ListQ[res],{fs/.res[[2]],Thread[vars->(vars/.res[[2]])]}]
];

(* syntax GoalAttainment[{{{f1,goal1,weight1},...},cons},vars]]*)
GoalProgrammingChebyshev[{fg:{{_,_}..},cons_},vars_,opts___?OptionQ]:=With[
{res=Catch[iGoalProgrammingChebyshev[{Map[(Append@@#&),Thread[{fg,ConstantArray[1,{Length[fg]}]}]],cons},vars]]},
res/;ListQ[res]
];
GoalProgrammingChebyshev[{fg:{{_,_,_}..},cons_},vars_,opts___?OptionQ]:=With[
{res=Catch[iGoalProgrammingChebyshev[{fg,cons},vars]]},
res/;ListQ[res]
];
iGoalProgrammingChebyshev[{fg:{{_,_,_}..},cons_},vars_,opts___?OptionQ]:=Module[
{fs,goals,y,res,ws},
{fs,goals,ws} = Transpose[fg];
If[!VectorQ[ws,(#>=0&)],Throw[$Failed]];
If[!VectorQ[goals,((NumericQ[#]&&Head[#]=!=Complex)&)],Throw[$Failed]];
res=FindMinimum[{y,(And@@Flatten[{cons},1])&&(And@@Thread[y≥ws*(fs-goals) ])},Append[Flatten[{vars},1],y],opts];
If[ListQ[res],{fs/.res[[2]],Thread[vars->(vars/.res[[2]])]}]
];

res1=GoalProgrammingWeightedAverage[{{{x^2+y^2,0},{4(x-2)^2+4(y-2)^2,0}},y-x==-4},{x,y}]

res2=GoalProgrammingChebyshev[{{{x^2+y^2,0},{4(x-2)^2+4(y-2)^2,0}},y-x==-4},{x,y}]

g1=ContourPlot[x^2+y^2,{x,2,6},{y,-1,2},ContourShading->False,ContourStyle->Blue,ContourLabels->Automatic];
g2=ContourPlot[4(x-2)^2+4(y-2)^2,{x,2,6},{y,-1,2},ContourShading->False,ContourStyle->Red,ContourLabels->Automatic];
Show[{g1,g2},Epilog->{Line[{{3,-1},{6,2}}],PointSize[0.02],Yellow,Point[{x,y}/.res1[[2]]],Green,Point[{x,y}/.res2[[2]]]}]

R = {{1.075`,1.084`,1.061`,1.052`,1.055`,1.077`,1.109`,1.127`,1.156`,1.117`,1.092`,1.103`,1.08`,1.063`,1.061`,1.071`,1.087`,1.08`,1.057`,1.036`,1.031`,1.045`},{0.942`,1.02`,1.056`,1.175`,1.002`,0.982`,0.978`,0.947`,1.003`,1.465`,0.985`,1.159`,1.366`,1.309`,0.925`,1.086`,1.212`,1.054`,1.193`,1.079`,1.217`,0.889`},{0.852`,0.735`,1.371`,1.236`,0.926`,1.064`,1.184`,1.323`,0.949`,1.215`,1.224`,1.061`,1.316`,1.186`,1.052`,1.165`,1.316`,0.968`,1.304`,1.076`,1.1`,1.012`},{0.815`,0.716`,1.385`,1.266`,0.974`,1.093`,1.256`,1.337`,0.963`,1.187`,1.235`,1.03`,1.326`,1.161`,1.023`,1.179`,1.292`,0.938`,1.342`,1.09`,1.113`,0.999`},{0.698`,0.662`,1.318`,1.28`,1.093`,1.146`,1.307`,1.367`,0.99`,1.213`,1.217`,0.903`,1.333`,1.086`,0.959`,1.165`,1.204`,0.83`,1.594`,1.174`,1.162`,0.968`},{1.023`,1.002`,0.123`,1.156`,1.03`,1.012`,1.023`,1.031`,1.073`,1.311`,1.08`,1.15`,1.213`,1.156`,1.023`,1.076`,1.142`,1.083`,1.161`,1.076`,1.11`,0.965`},{0.851`,0.768`,1.354`,1.025`,1.181`,1.326`,1.048`,1.226`,0.977`,0.981`,1.237`,1.074`,1.562`,1.694`,1.246`,1.283`,1.105`,0.766`,1.121`,0.878`,1.326`,1.078`},{1.677`,1.722`,0.76`,0.96`,1.2`,1.295`,2.212`,1.296`,0.688`,1.084`,0.872`,0.825`,1.006`,1.216`,1.244`,0.861`,0.977`,0.922`,0.958`,0.926`,1.146`,0.99`}};

{n,nyear}=Dimensions[R];

ER=Mean[Transpose@R]

Covariants=Covariance[Transpose[R]];

vars = Map[Subscript[x,#]&,{"3m T-bill","long T-bond","SP500","Wiltshire 5000","Corporate Bond","NASDQ","EAFE","Gold"}];
vars = Map[Subscript[x,#]&,{"3m T-bill","long T-bond","SP500","Wiltshire 5000","Corporate Bond","NASDQ","EAFE","Gold"}];
FindMinimum[{
vars.Covariants.vars,Total[vars]==1&&vars.ER≥1.12&&Apply[And,Thread[Greater[vars,0]]]},vars]

vars = Map[Subscript[x,#]&,{"3m T-bill","long T-bond","SP500","Wiltshire 5000","Corporate Bond","NASDQ","EAFE","Gold"}];
FindMinimum[{
vars.Covariants.vars,Total[vars]==1&&vars.ER≥1.10&&Apply[And,Thread[Greater[vars,0]]]},vars]

FindMinimum[{Abs[x^2-3],0≤x≤5},{x},Method->InteriorPoint]

FindMinimum[{Abs[x^2-3]},{x},Method->Newton]

Minf(x)
s.t.h(x)==0,x≥0.

g(x)-z-(y^T)A(x)==0
h(x)==0
ZXe==μ e.

(({{H(x,y)+(X^(-1))Z, -((A(x))^T)}, {-A(x), 0}}))(({{δx}, {δy}}))==-(({{(Subscript[d, ψ])+(X^(-1))(Subscript[d, xz])}, {-(Subscript[d, h])}}))==-(({{g(x)-((A(x))^T)y-μ(X^(-1))e}, {-h(x)}})).

x:=x+δx,y:=y+δy,z:=z+δz

Augmented Lagrangian Merit Function

ϕ(x,β)==f(x)-μ(∑_i)ln((Subscript[x, i]))-h(((x))^T)λ+β||h(x)((||)^2).

Convergence Tolerance