From 25fdd4e598aa337bf8643fe42e8cab820ebcc6f8 Mon Sep 17 00:00:00 2001 From: Claudio Maggioni Date: Wed, 26 May 2021 19:13:56 +0200 Subject: [PATCH] hw4 is done --- Claudio_Maggioni_4/Claudio_Maggioni_4.md | 70 +++++++++++++++++++++- Claudio_Maggioni_4/Claudio_Maggioni_4.pdf | Bin 150277 -> 154515 bytes 2 files changed, 68 insertions(+), 2 deletions(-) diff --git a/Claudio_Maggioni_4/Claudio_Maggioni_4.md b/Claudio_Maggioni_4/Claudio_Maggioni_4.md index eafebb0..0efa064 100644 --- a/Claudio_Maggioni_4/Claudio_Maggioni_4.md +++ b/Claudio_Maggioni_4/Claudio_Maggioni_4.md @@ -36,7 +36,7 @@ $$\nabla_X L(X,\lambda) = \begin{bmatrix}(-3-\lambda)x^* + 1\\(1-\lambda)y^* + \begin{bmatrix}x^*\\y^*\\z^*\end{bmatrix} = \begin{bmatrix}\frac1{3+\lambda}\\-\frac1{1-\lambda}\\-\frac{1}{2-\lambda}\end{bmatrix}$$ -Then we have the conditions on the equality constraint: +Then we have the complementarity condition: $$c(X) = {x^*}^2 + {y^*}^2 + {z^*}^2 - 1 = 0 \Leftrightarrow \|X^*\| = 1$$ @@ -141,10 +141,76 @@ Then we have the conditions on the equality constraint: $$Ax - b = 0 \Leftrightarrow \begin{bmatrix}x_1 + x_3\\x_2 + x_3\end{bmatrix} = \begin{bmatrix}3\\0\end{bmatrix}$$ -Then we have the conditions on the equality constraint: +Then we have the complementarity condition: $$\lambda^T (Ax - b) = 0 \Leftarrow Ax - b = 0 \text{ which is true if the above condition is true.}$$ Since we have no inequality constraints, we don't need to apply the KKT conditions realated to inequality constraints. + +# Exercise 3 + +## Exercise 3.1 + +The lagrangian of this problem is the following: + +$$L(x, \lambda) = c^T x - \lambda^T (Ax - b) - s^T x$$ + +The KKT conditions are the following: + +1. The partial derivative of the lagrangian w.r.t. $x$ is 0: + $$\nabla_x L(x, \lambda) = c - A^T \lambda - s = 0 \Leftrightarrow A^T \lambda + + s = c$$ +2. Equality constraints hold: + $$Ax - b = 0 \Leftrightarrow Ax = b$$ +3. Inequality constraints hold: + $$x \geq 0$$ +4. The lagrangian multipliers for inequality constraints are non-negative: + $$s \geq 0$$ +5. The complementarity condition holds (here considering only inequality constraints, + since the condition trivially holds for equality ones): + $$s^T x \geq 0$$ + +## Exercise 3.2 + +We define the dual problem is the following way: + +$$\max b^T \lambda \;\; \text{ s.t. } \;\; A^T \lambda + s = c \; \text{ and +}\; s \geq 0$$ + +To convert this maximization problem in a minimization one, we flip the sign of +the objective and we find: + +$$\min - b^T \lambda \;\; \text{ s.t. } \;\; A^T \lambda + s = c \; \text{ and +}\; s \geq 0$$ + +We then compute the Lagrangian of the dual problem: + +$$L(\lambda, x) = -b^T \lambda + x^T (A^T \lambda + s - c) - x^T s = - b^T \lambda + x^T (A^T \lambda - c)$$ + +The KKT conditions are the following: + +1. The partial derivative of the lagrangian w.r.t. $x$ is 0: + $$\nabla_{\lambda} L(\lambda, x) = - b^T + x^T A^T = 0 \Leftrightarrow Ax = b$$ +2. Equality constraints hold: + $$A^T \lambda + s = c$$ +3. Inequality constraints hold: + $$c - A^T \lambda \geq 0 \Leftrightarrow s \geq 0 \;\;\; \text{ using 2.\ to find + that } s = c - A^T \lambda$$ +4. The lagrangian multipliers for inequality constraints are non-negative: + $$x \geq 0$$ +5. The complementarity condition holds (here considering only inequality constraints, + since the condition trivially holds for equality ones): + $$x^T s \geq 0 \Leftrightarrow s^T x \geq 0$$ + +Then, if we compare the KKT conditions of the primal problem with the ones above +we can match them to see that they are identical: + +- 1.\ from the dual is identical to 2.\ from the primal; +- 2.\ from the dual is identical to 1.\ from the primal; +- 3.\ from the dual is identical to 4.\ from the primal; +- 4.\ from the dual is identical to 3.\ from the primal; +- 5.\ from the dual is identical to 5.\ from the primal. + +Therefore, the primal and the dual problem are equivalent. diff --git a/Claudio_Maggioni_4/Claudio_Maggioni_4.pdf b/Claudio_Maggioni_4/Claudio_Maggioni_4.pdf index fd72b342b6074c6c46d0f4c5239d81d10473c507..fcbd8e370bded5a795c0b0283c2b52fbdb813f0e 100644 GIT binary patch delta 53752 zcmZsBQ*b5@@MLT|8)IYJ=Ek;d8((bOwvA1)v2EM7@!juVS6BCNH&6XAFH<#D-P1jx zX|T!dunE6Gxzd@r!KeV&x;ieK99aIZ^>drb$#C%aTbt*THzi%ob`}(jf<|rF8p-=IN`=`N;r>n{)*i#c}gbqknrCX#0XYDhYs}O#so$y59@)wncVp z*FX|c`e)a)FDOsv&NQI&3)vZsbtEDc4WE(>cG0_SF9^qszZg)rFnWn_doXr2`t*WT zY4&3d$w~o(P1EbuAM;x~dR+v^IOCzFy7a zvDQ;M4TGsuq>+iCzWXUm=w=8d-6(i^zrC$2VOAa@CLPYoNAb?{Wi?hqxP)f!-kYC z2K9|b1VS$~i7-73JyMRJF#l};NUAV>wahXljsP?2x zSq^0+!~jBRS3S&xN~3z`Y5j?N0v;k~zy=f~e7JD2*k7bO!avU63u%B4W_K6(Fz zb12=2bcY?0kw#}3Q3EHFN7X>i8s~*WOX!UoKB(YDQ;E;f*OzKODL+O5ib==(jK(sh zRTO81gU@U3plly$D=>z{7hxN&LPrRq7xf5Rl@3@;M4VGXn#?G_fU)BlIGwvUQI$+D zm0!M}O)GRMHEzK%w;qo}L~9$+bta`cl4}v)4B5VW-8J}ld9|KNn*QK%SrnQ_E#EX& zlCl~I*xM6SueCg{`pOZdyDV+CS;^-iKGI&YiQr z1pw(R{rkh{TbLd@Y+3I_zqWtS*OT}oxl4sSF>GA_&TX9g6GRq45~Gf6M~FsJHGQ=i znp-2qnM5|jkbxyz9( zOFy{b0%%R-8QC#FFd;!Al-odr)Is_vLGF~8yN0Fio94_7wRm6 z(|0GX|C$XWDwN`3DSL!Aph#ca|1%tO|SJDkv`4rQHU?$1U zcaG$bJ45a?XN`S?`WP!(0~cUhyiB~l`5v_P>K&4B=ZDU{TC{8PfY1LPXaW$#T???9 zPV4@ub`Vr(n&r0j@Yr*A@mW>LUbB|wxPv)3GczLW&-?c_D@ihB}$ddsO%6!G)IiQK;5#}omx|U9y`P1`zWRU zRpR0}y&w3!sONs{e5ZG)>G-Kxa(h+QNSpW*RU+3!{m?CN^gd!Qp?)RD_2+#CJnkta zvM9+q>yrrQ8rk7iDaB*`V`APJj~@E3p%P14$<1vNE`z4SSNyv7`zFBTZ`B2izNoPK zr(IUIEufW7`R&`g`DgPVaf`10XY^-mZ;}9C4cs4ni|&)xXg<4g(|zmLzp+L|TER~9nzBt=q*s)ry>1ZZe#XE!fRuLivbx$|VxI?9gu`8rPR9Vmm76_B)=gVNZ z*$}R^A@-?vZ8={F7BpR98G?9unpk7d!$X zd?qzV_3|{EwWbU8h8+7?F}kDdFoY0-P|l<;cmu(Gk}837dIztv+9Q|W5|&Q~!4VDU zUT@jPw4g6!w(q<84<_uYFuO^FS@fM$>mC&XRcJ z0oKD88~-RLTFytz20q`*)i@;0LUu%Yhrw@YxI0+yc(A<`70f$!BpP;X@!5 z4!syS5&FC=`gpwHZZhXq9=Xu76If_o(AD4qRs*PR3U(Z9$YrLp{d2);OA0T#te&e7 z@fuwhPaghgiRdk6fuAj*=8!eQpLn26aXr;Yg~WD&Tpf~gVS$@(y>Whb?p4rzH*q(2 zC`7dbVZw7?lbAianrS1#fCFYJ_kwy?QK!yUQ$_!%TW%d-=fZc4*Z&b~zO~bjO z695Ei*(z$D3CdOfUe25n4Vi)#e&{P+p-_7|B*aSx(Y-uDwY6X!Y5rS0hJvE}TSkIm zeVgNnsP?^#`n6Y!WEh!vC5D`F!9Ja+o(E>ffq%`o0RG196=3-oFkflGgB(z)JWI(7 z^27ro*bjPD_@^`63?UJJpuJ_Mg?}wng-iCXpZJbAu|2R)GJrTNc#b|9=W)dl<1>*x z5V=a6#K4I15I%XmF9MdJaY`(_Ip#r-)k;8*{=Ul%*c-t+4nsI3xOM;XZ~%l9k0fXg zf+K~2U}5Iq{NLJgTl@cL%RQZA>4*dWb9oOA-egyAuX9yo-sJ0GQfdwh(}~iQjrUi6 zVj;8up~xfx&P8mUpuXT>mv4c9jt*+a`$5bXAdt8B<0ZEQl4J}ifAAGtxXuINZBg5I@0mjAGu&$fD{*qO-E~>Wcd&;Ir zGZ)|(2Sfh{Dtmv*>MlB)QdI7@@-O$@pC167qXdSicBK8xVQ-xbDdtouDOnthDGoeD z(j_2?hW~SChQ5XyqTR-sGjMIcy8W?QunvYV1YTLuC?TONA&DfRrMt@;$jhL9x2XWB zxqNm|nCqqCXxidut@7pmT>YwFA;T!`7}DrU>+N8LzaG#R_IqIp@4|fuj<0n9IUd(3 zg=AcgvvuMZ!R5|HJqzho`A$@FtCl%AY8>EgF9EJwnEDO!6kkq=Q5fpfjGrzctb>Pu zxyRkXnInkHNO;R^*-FsOMZi>bKZ1rE%jS0~l$E_o0O=y+)~h^%sYe04L{NnYS&g@! 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