also has x æ as a local imizer. Of course, æ is typically not known, but an algorithm can approximate æ as it approximates x æ èas the augmented Lagra
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1 Introduction to sequential quadratic programg Mark S. Gockenbach Introduction Sequential quadratic programg èsqpè methods attempt to solve a nonlinear program directly rather than convert it to a sequence of unconstrained imization problems. To make this introduction as simple as possible, I will begin by discussing the SQP framework for equality-constrained NLPs. The basic idea is analogous to Newton's method for unconstrained imization: At each step, a local model of the optimization problem is constructed and solved, yielding a step èhopefullyè toward the solution of the original problem. In unconstrained imization, only the objective function must be approximated, and the local model is quadratic. In the NLP s:t: gèxè =; both the objective function and the constraint must be modeled. An SQP method uses a quadratic model for the objective function and a linear model of the constraint. A nonlinear program in which the objective function is quadratic and the constraints are linear is called a quadratic program èqpè. An SQP method solves a QP at each iteration. Given an current estimate x èkè of a solution x æ, g can be approximated by gèx èkè + pè : = rgèx èkè è T p + gèx èkè è; and so the constraint is replaced by gèxè = rgèx èkè è T p + gèx èkè è=: At ærst glance, one would expect that the quadratic objective function for the model problem would be the Taylor approximation to f: fèx èkè + pè : = fèx èkè è+rfèx èkè è æ p + p ær fèx èkè èp: However, this would be the wrong choice, because the curvature of the constraints must be captured by the model problem. I demonstrate this by an example below. If æ is the Lagrange multiplier corresponding to a local imizer x æ of s:t: gèxè =; then the Lagrangian `èæ; æ è has the property that `èx; æ è= for all feasible x. It follows that `èx; æ è s:t: gèxè =
2 also has x æ as a local imizer. Of course, æ is typically not known, but an algorithm can approximate æ as it approximates x æ èas the augmented Lagrangian method does, for exampleè. Given x èkè and èkè, `èx èkè + p; èkè è : = p ær`èx èkè ; èkè èp + r`èx èkè ; èkè è æ p + `èx èkè ; èkè è èfor p near è: I will show below that solving p ær`èx èkè ; èkè èp + r`èx èkè ; èkè è+`èx èkè ; èkè è s:t: rgèx èkè è T p + gèx èkè è= yields improved values of x èkè and èkè èat least when x èkè and èkè are close to x æ and æ, respectivelyè. First, however, I give an example that illustrates the necessity of using the Lagrangian to deæne the quadratic programg subproblems. Example. Deæne f :R! R and g :R! R by Then the solution of the nonlinear program = èx, è, x ; gèxè = x + x, : s:t: gèxè = is x æ = è; è, and the corresponding Lagrange multiplier is æ =,, as can be easily checked. Taking x èè =è=; =è, I form the QP or s:t: p ær fèx èè èp + rfèx èè è æ p + fèx èè è rgèx èè è T p + gèx èè è=;,p + p, p, p + s:t: p +p + 9 =; in hopes that the solution p èè will deæne a step towards x æ, allowing me to deæne x èè = x èè + p èè. However, the quadratic program is unbounded below and hence has no solution. This is clearly seen in Figure, which shows the contours of the quadratic objective function together with the linearized constraint and the original nonlinear constraint. This example shows that the quadratic objective function in the QP subproblem cannot simply approximate f. By contrast, if èè =,=, then simpliæes to p ær`èx èè ; èè èp + r`èx èè ; èè è æ p + `èx èè ; èè è s:t: rgèx èè è T p + gèx èè è= p +p + p + p + 77 s:t: p +p + 9 =: This QP is well-posed; it is illustrated in Figure,which also shows x èè = x èè + p èè, where p èè is the solution to the QP. As Figure shows, p èè isagood step towards x æ.
3 Decreasing Decreasing Figure : An illustration of the ærst proposed quadratic program in Example.. The nonlinear constraint, the linearized constraint, and the contours of the quadratic objective function are shown. The asterisk indicates x èè =è=; =è. Figure : An illustration of the second proposed quadratic program in Example.. The nonlinear constraint, the linearized constraint, and the contours of the quadratic objective function are shown. The asterisk indicates x èè =è=; =è, while the small circle indicates x èè =è,:8; : :9è. Relationship of Newton's method to SQP If x æ is a local imizer and nonsingular point of
4 s:t: gèxè =; and æ is the corresponding Lagrange multiplier, then rfèx æ è,rgèx æ è æ = ;,gèx æ è = : r `èx æ ; æ è,rgèx æ è,rgèx æ è T is nonsingular. It is therefore reasonable to try to compute x æ ; æ by applying Newton's method to the system r,rgèxè = ;,gèxè = : Given an estimate èx èkè ; èkè èofèx æ ; æ è, Newton's method deænes èx èk+è ; èk+è è=èx èkè ; èkè è+ èp èkè ;! èkè è, where the step èp èkè ;! èkè è is deæned by the linear system r `èx èkè ; èkè è,rgèx èkè è,rgèx èkè è T p èkè! èkè =, r`èx èkè ; èkè è,gèx èkè è : èè On the other hand, assug èx èkè ; èkè è is suæciently close to èx æ ; æ è that r `èx èkè ; èkè è is positive deænite on the null space of rgèx èkè è T, then the quadratic program p ær`èx èkè ; èkè èp + r`èx èkè ; èkè è+`èx èkè ; èkè è s:t: rgèx èkè è T p + gèx èkè è= is a convex program and has a unique solution-lagrange multiplier pair èp èkè ;! èkè è. This pair is detered by the ærst-order necessary conditions: r `èx èkè ; èkè èp èkè + r`èx èkè ; èkè è,rgèx èkè è! èkè = ; èè rgèx èkè è T p èkè + gèx èkè è = : èè But the system èíè is the same as system èè. In other words, the SQP method is equivalent to Newton's method applied to the ærst-order necessary conditions!. Therefore, at least locally, the SQP method deænes not only a good step from x èkè towards x æ, but also a good step from èkè towards æ. In fact, under the usual assumptions, èx èkè ; èkè è! èx æ ; æ è quadratically. The advantage of the SQP framework over simply applying Newton's method to the ærst-order necessary conditions is that the optimization framework gives us some basis for modifying the step when èx èkè ; èkè è is not suæciently close to èx æ ; æ è that pure Newton's method deænes a good step. This is the same reason that, in unconstrained imization, it is advantageous to think of Newton's method as repeatedly imizing a quadratic model rather than as trying to ænd a zero of the gradient. Sequential quadratic programg deænes a locally convergent algorithm, and it is in this sense that one can talk about the SQP method. However, diæerent techniques can be used to create a globally-convergent iteration, and so there can be many SQP algorithms. Example. Figure illustrates four iterations of the local SQP method applied to Example. with x èè =è; è and èè ==. The numerical results of nine iterations are summarized intable, which shows x èkè! x æ =è; è and èkè! æ =,.
5 Figure : Four iterations of the local SQP method applied to the NLP from Example.. The nonlinear constraint, the linearized constraint, and the contours of the quadratic objective function are shown. The asterisk indicates x èkè and the small circle x èk+è, k = èupper leftè, k = èupper rightè, k = èlower leftè, and k = èlower rightè. An alternate formulation of the local SQP method The following calculation suggests a slightly diæerent formulation of the SQP algorithm: r`èx èkè ; èkè è æ p = The last equality is valid if p satisæes the constraint rfèx èkè è,rgèx èkè è èkè æ p = rfèx èkè è æ p, rgèx èkè è èkè æ p = rfèx èkè è æ p, èkè æ rgèx èkè è T p = rfèx èkè è æ p + èkè æ gèx èkè è rgèx èkè è T p + gèx èkè è=: èè It follows that, on the linearized feasible set deæned by èè, the two quadratics and p ær`èx èkè ; èkè èp + r`èx èkè ; èkè è+`èx èkè ; èkè è p ær`èx èkè ; èkè èp + rfèx èkè è+fèx èkè è
6 k x èkè x èkè èkè : :,: :9 :7,: : æ, :8,:,:8 æ, :7,:7 :78 æ, :,:879,:9 æ, :98,:877 :89 æ, :9,:88 7,:9 æ, :,:997 8 :8 æ, :,: 9,:79 æ, :,: Table : Nine iterations of the SQP method applied to Example.. See also Figure. diæer by a constant, and therefore the QPs and p ær`èx èkè ; èkè èp + r`èx èkè ; èkè è+`èx èkè ; èkè è èè s:t: rgèx èkè è T p + gèx èkè è= èè p ær`èx èkè ; èkè èp + rfèx èkè è+fèx èkè è è7è s:t: rgèx èkè è T p + gèx èkè è= è8è have the same solution p èkè. However, the Lagrange multipliers for the two QPs diæer since the gradient of the objective functions are diæerent. The optimality conditions for è7í8è are or, equivalently, r `èx èkè ; èkè èp èkè + rfèx èkè è,rgèx èkè è! èkè = ; rgèx èkè è T p èkè + gèx èkè è = ; r `èx èkè ; èkè èp èkè,rgèx èkè è! èkè =,rfèx èkè è; rgèx èkè è T p èkè + gèx èkè è = : Adding rgèx èkè è èkè to both sides of the ærst equation yields r `èx èkè ; èkè èp èkè,rgèx èkè è! èkè, èkè =,r`èx èkè ; èkè è; è9è rgèx èkè è T p èkè + gèx èkè è = : èè I have already observed that QPs èíè and è7í8è have the same solution p èkè. optimality conditions èíè and è9íè shows that Comparing the! èkè, èkè =! èkè ; that is,! èkè = èkè +! èkè = èk+è : Therefore, when the QP is formulated as è7í8è, the Lagrange multiplier for the QP is not the step to èk+è, it is actually èk+è itself. For equality-constrained NLPs, this is the only signiæcant
7 diæerence between the two formulations of the QP subproblem. However, as I will show, the diæerence is much more signiæcant for inequality-constrained NLPs, and it is necessary to adopt version quadratic objective è7è. A ænal remark is that the constant term in the quadratic objective is irrelevant for detering p èkè and! èkè or! èkè and so it is usually not included. Therefore the objective function for the QP is taken to be p ær`èx èkè ; èkè èp + r`èx èkè ; èkè è or p ær`èx èkè ; èkè èp + rfèx èkè è 7
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