So far we learned a lot about the structure of LPs. But how do we actually solve them? Can we do it faster than by just enumerating all basic feasible solutions (BFS)?

• Simplex – 1940s.

• Ellipsoid method – 1970s.

• Interior point method – 1980s.

Simplex

• Recall that we know that for an LP (in standard form) opt always corresponds to a vertex of a polytope (a BFS).

• Imagine we are given some BFS, how to check if it is optimal?

• Naive approach: check all other BFS’ and compare their costs.

  \(\Rightarrow\) Problem: there can be \(m \choose n\) of them!

• But do we really need to look at all the other BFS’?

Interlude: Vertices in Standard form/bases

• Without loss of generality make \(A\) have full row rank (define):

  • find basis in rows of \(A\), say \(a_1,\ldots,a_k\)

  • any other \(a_\ell\) is linear combo of those.

  • so \(a_\ell x = \sum \lambda_i a_i x\)

  • so better have \(b_l = \sum \lambda_i a_i\) if any solution.

  • if so, anything feasible for \(a_1,\ldots,a_\ell\) feasible for all.

• \(m\) constraints \(Ax=b\) all tight/active

• given this, need \(n-m\) of the \(x_i \ge 0\) constraints

• also, need them to form a basis with the \(a_i\).

• write matrix of tight constraints, first \(m\) rows then identity matrix adjacent to zero matrix \[\begin{array}{l} \\ \\ m\text{ constraint rows} \\ \\ \\ \\ \\ \\ n\text{ row identity} \\ \\ \\ \end{array} \left( \begin{array}{lccccccc} &\multicolumn{4}{c}{n-m\text{ columns}}&\multicolumn{3}{c}{m\text{ columns}}\\ &\multicolumn{4}{c}{\overbrace{\qquad\qquad\qquad}}&\multicolumn{3}{c}{\overbrace{\qquad\qquad}}\\ \\ &\multicolumn{3}{c}{\cdots}&A&\multicolumn{3}{c}{\cdots}\\ \\ &1&\\ &&1&&&&0\\ &&&1\\ &&&&1\\ &&&&&1\\ &&0&&&&1\\ &&&&&&&1\\ \end{array} \right)x \begin{array}{c} \\ \\ = \\ \\ \\ \\ \ge \\ \\ \\ = \\ \end{array} \left( \begin{array}{c} \\ \\ \\ b \\ \\ \\ \\ 0 \\ \\ \\ 0 \\ \\ \end{array} \right)\]

• zero matrix corresponds to slack (nonzero) \(x_i\)

• need linearly independent rows

• equiv, need linearly independent columns

• but columns are linearly independent iff \(m\) columns of \(A\) including all that correspond to nonzero \(x_i\) are linearly independent

  • because columns of identity matrix are clearly independent

  • but columns of zero matrix depend entirely on \(A\) for independence

• gives other way to define a vertex: \(x\) is vertex if

  • \(Ax=b\)

  • \(m\) linearly independent columns of \(A\) include all \(x_j \ne 0\)

  This set of \(m\) columns is called a basis.

• \(x_j\) of columns called basic set \(B\), others nonbasic set \(N\)

• given bases, can compute \(x\):

  • \(A_B\) is basis columns, \(m \times m\) and full rank.

  • solve \(A_B x_B = b\), set other \(x_N=0\).

  • note can have many bases (column sets) for same vertex (choice of 0 \(x_j\))

  • note algebra is \(m\)-dimensional, so really depends only on number of constraints not variables

Summary: \(x\) is vertex of \(P\) if for some basis \(B\),

• \(x_N=0\)

• \(A_B\) nonsingular

• \(A_B^{-1} b \ge 0\)

Back to the Simplex Method

• The algorithm:

  • start with a basic feasible soluion

  • try to improve it

  • picture: move on improving edge.

• math: work relative to current \(x\)

• rewrite LP: \(\min c_Bx_B + c_N x_N\), \(A_Bx_B+A_N x_N=b\), \(x \ge 0\)

• true for all \(x\), not just current

• \(B\) is basis for bfs

• since \(A_Bx_B = b-A_Nx_N\), so \(x_B = A_B^{-1}(b-A_Nx_N)\), know that \[\begin{aligned} cx &= &c_Bx_B+c_Nx_N\\ &= & c_B A_B^{-1}(b-A_Nx_N) + c_N x_N\\ &= & c_B A_B^{-1} b + (c_N-c_BA_B^{-1}A_N)x_N\\ \end{aligned}\]

• reduced cost \(\tilde c_N = c_N-c_BA_B^{-1}A_N\)

• if no \(\tilde c_j < 0\), then increasing any \(x_j \in N\) increases (nondecreases) cost (may violate feasiblity for \(x_B\), but who cares?), so are at optimum!

• if some \(\tilde c_j < 0\), can increase \(x_j\) to decrease cost

• but since \(x_B\) is func of \(x_N\), will have to stop when \(x_B\) hits a constraint.

  • this happens when some \(x_i\), \(i \in B\) hits 0.

  • we bring \(j\) into basis, take \(i\) out of basis.

  • we’ve moved to an adjacent basis.

  • called a pivot

• show picture

Potential problems

• Need initial vertex. How find? HW.

• maybe some \(x_i \in B\) already 0, so can’t increase \(x_j\), just pivot to same obj value.

• could lead to cycle in pivoting, infinite loop.

• can prove exist noncycling pivots (eg, lexicographically first \(j\) and \(i\))

• Would hope that you don’t need to visit too many of the BFS. Not true! Klee-Minty cube (a slightly twisted hypercube)

• We can’t prove though that there is no pivot choice rule that leads to always fast simplex (the best ones are “only” subexponential)

• Hirch conjecture (1957): diameter/number of pivots needed is at most \(m+n\) (very optimistic!) Disproved by Santos in 2010 but only barely so: diameter \(\geq (1+\eps)(m+n)\).

• Still possible that, e.g., diameter \(\leq poly(m,n)\).

• Kalai-Kleitmen 1992: \(m^{\log n}\) bound on path length, also \(2^n m\).

• There is a linear time simplex for fixed \(n\)!

• Note small diameter does not guarantee fast simplex pivot rule - as some of the pivots might require increasing the value of the objective from time to time!

• In practice: works really really well (unless it doesn’t). (Needs some linear algebra tricks to make pivoting fast.)

Simplex and Duality

• defined reduced costs of nonbasic vars \(N\) by \[{\tilde c}_N = c_N-c_BA^{-1}_BA_N\] and argued that when all \({\tilde c}_N \ge 0\), had optimum.

• Define \(y=c_BA^{-1}_B\) (so of course \(c_B=yA_B\))

• nonegative reduced costs means \(c_N \ge yA_N\)

• put together with \(c_B=yA_B\), see \(yA \le c\) so \(y\) is dual feasible

• but, \(yb = c_B A^{-1}_B b = c_Bx_B=cx\) (since \(x_N=0\))

• so \(y\) is dual optimum.

• simplex finds primal and dual optima simultaneously

• more generally, \(yb-cx\) measures duality gap for current solution!

• another way to prove duality theorem: prove there is a terminating (non cycling) simplex algorithm.

Polynomial Time Bounds

We know a lot about structure. And we’ve seen how to verify optimality in polynomial time. Now turn to question: can we solve in polynomial time?

Yes, sort of (Khachiyan 1979):

• polynomial algorithms exist

• strongly polynomial unknown.

Ellipsoid Method

Basic idea: Reduce optimization to feasibility checking. (Remind how it is with polytope \(P\).)

• Bolzano-Weierstrass Theorem—proves certain sequence has a subsequence with a limit by repeated subdividing of intervals to get a point in the subinterval.

• The Bolzano-Weierstrass method: Divide the desert by a line running from north to south. The lion is then either in the eastern or in the western part. Let’s assume it is in the eastern part. Divide this part by a line running from east to west. The lion is either in the northern or in the southern part. Let’s assume it is in the northern part. We can continue this process arbitrarily and thereby constructing with each step an increasingly narrow fence around the selected area. The diameter of the chosen partitions converges to zero so that the lion is caged into a fence of arbitrarily small diameter.

• Analogue of a fence: separation oracle. Given point \(x\) that is not in \(P\) come up with a separating hyperplane, i.e., \(u\) such that \(u^T x \leq 0\) but \(y^Tx' > 0\) for all \(x'\in P\).

• We want to use ellipsoid instead of boxes.

• Ellipsoid \(E(D,z):=\{x \mid (x-z)^TD^{-1}(x-z) \leq 1 \}\), where \(D=BB^T\) for some invertible matrix \(B\).

• Note: \(E(r\cdot I,z)\) is just an radius \(r\) sphere centered at \(z\).

• In general: \(E(D,z)\) is a mapping of a unit sphere via an affine change of coordinates \(x\rightarrow Bx +z\).

• Goal: start with a big ellipsoid that encompasses everything (for sure).

• In each step, query the center of the ellipsoid \(z\). If \(z\in P\) done. Otherwise, there is a separating hyperplane \(u\). Shift it so it passes through the center and draw a smaller ellipsoid that encompasses the half in which the feasible point might lie.

• How to measure progress? Keep track of the volume of the ellipsoid. Want it to shrink significantly in each step.

• Outline of algorithm:

  • solve feasibility, which we know is equivalent to optimizing

  • goal: find a feasible point for \(P=\set{Ax \le b}\)

  • start with ellipse containing \(P\), center \(z\)

  • check if \(z \in P\)

  • if not, use separating hyperplane to get 1/2 of ellipse containing \(P\)

  • find a smaller ellipse containing this 1/2 of original ellipse

  • until center of ellipse is in \(P\).

• Consider sphere case (wlog as affine transformations change the volume by the same factor and we care about the ratio change), separating hyperplane \(x_1=0\)

  • try center at \((\epsilon,0,0,\ldots)\)

  • Draw picture to see constraints

  • requirements:

    • \(d_1^{-1}(x_1-\epsilon )^2+\sum_{i > 1}d_i^{-1}x_i^2 \le 1\)

    • constraint at \((1,0,0)\): \(d_1^{-1}(x-\epsilon )^2 = 1\) so \(d_1 = (1-\epsilon )^2\)

    • constraint at \((0,1,0)\): \(\epsilon ^2/(1-\epsilon )^2+d_2^{-1} = 1\) so \(d_2^{-1} = 1-\epsilon ^2/(1-\epsilon )^2\approx 1-\epsilon ^2\)

  • What is volume? about \((1-\epsilon )/(1-\epsilon ^2)^{n/2}\)

  • set \(\epsilon\) about \(1/n\), get \((1-1/n)\) volume ratio.

• Formalize shrinking lemma:

  • Let \(E=(z,D)\) define an \(n\)-dimensional ellipsoid

  • consider separating hyperplane \(ax \le az\)

  • Define \(E'=(z',D')\) ellipsoid: \[\begin{aligned} z' &= &z-\frac{1}{n+1}\frac{Da^T}{\sqrt{aDa^T}}\\ D' &= & \frac{n^2}{n^2-1}(D-\frac{2}{n+1}\frac{Da^TaD}{aDa^T}) \end{aligned}\]

  • then \[\begin{aligned} E \cap \set{x \mid ax \le ez} &\subseteq &E'\\ {\mbox{vol}}(E') &\le &e^{1/(2n+1)}{\mbox{vol}}(E) \end{aligned}\]

  • for proof, first show works with \(D=I\) and \(z=0\). new ellipse: \[\begin{aligned} z' &=&-\frac1{n+1}\\ D' &= & \frac{n^2}{n^2-1}(I-\frac{2}{n+1} I_{11}) \end{aligned}\] and volume ratio easy to compute directly.

  • for general case, transform to coordinates where \(D=I\) (using new basis \(B\)), get new ellipse, transform back to old coordinates, get \((z',D')\) (note transformation don’t affect volume ratios.

• So ellipsoid shrinks. Now prove 2 things:

  • needn’t start infinitely large

  • can’t get infinitely small

• Starting size:

  • recall bounds on size of vertices (polynomial)

  • so coords of vertices are exponential but no larger

  • so can start with sphere with radius exceeding this exponential bound

  • this only uses polynomial values in \(D\) matrix.

  • if unbounded, no vertices of \(P\), will get vertex of box.

• Ending size:

  • suppose polytope full dimensional

  • if so, it has \(n+1\) affinely indpendent vertices

  • all the vertices have poly size coordinates

  • so they contain a box whose volume is a poly-size number (computable as determinant of vertex coordinates)

• Put together:

  • starting volume \(2^{n^{O(1)}}\)

  • ending volume \(2^{-n^{O(1)}}\)

  • each iteration reduces volume by \(e^{1/(2n+1)}\) factor

  • so \(2n+1\) iters reduce by \(e\)

  • so \(n^{O(1)}\) reduce by \(e^{n^{O(1)}}\)

  • at which point, ellipse doesn’t contain \(P\), contra

  • must have hit a point in \(P\) before.

• Justifying full dimensional:

  • feasible points form a “lattice” of grid points with bounded number of bits

  • take \(\set{Ax \le b}\), replace with \(P'=\set{Ax \le b+\epsilon}\) for tiny \(\epsilon\)

  • any point of \(P\) is an interior of \(P'\), so \(P'\) full dimensional (only have interior for full dimensional objects)

  • \(P\) empty (of lattice points) iff \(P'\) is (because \(\epsilon\) so small)

  • can “round” a point of \(P'\) to \(P\).

• Infinite precision:

  • built a new ellipsoid each time.

  • maybe its bits got big?

  • no.

• Optimization

  • Could use optimization to feasibility transform

  • But an easier approach is binary search on value of optimum

  • Add constraint that optimum must exceed proposed value

  • Vary value

• Need to find vertex?

  • use rounding technique

Feasibility vs. Separation vs. Optimization

Notice in ellipsoid, were only using one constraint at a time.

• didn’t matter how many there were.

• didn’t need to see all of them at once.

• just needed each to be represented in polynomial size.

• so ellipsoid works, even if huge number of constraints, so long as have separation oracle: given point not in \(P\), find separating hyperplane.

• of course, feasibility is same as optimize, so can optimize with sep oracle too.

• this is on a polytope by polytope basis. If can separate a particular polytope, can optimize over that polytope.

• Another good reason to optimize by binary search on objective:

  • just need feasibility oracle/separator

  • Might not have separator for combined primal-dual formulation

  • especially since dual can have exponentially many variables!

This is very useful in many applications. e.g. network design.

Can also show that optimization implies separation:

• suppose can optimize over \(P\)

• then of course can find a point in \(P\)

• suppose \(0\in P\) (saves notation mess—just shift \(P\))

• define \(P^* = \set{z \mid zx \le 1\ \forall x \in P}\)

• can separate over \(P^*\):

  • given \(w\), run OPT\((p)\) with \(w\) objective

  • get \(x^*\) maximizing \(wx\)

  • if \(wx^* \le 1\) then \(w \in P^*\)

  • else \(wx^* > 1 \ge x^*z \ \forall z \in P^*\) so \(x^*\) is separating hyperplane

  • since can separate \(P^*\), can optimize it

• suppose want to separate \(y\) from \(P\)

• let \(z=\)OPT\((P^*,y)\).

• if \(yz>1\) then (since \(z \in P^*\)) we have \(yz>1\) but \(xz \le 1 \ \forall x \in P\) (separating hyperplane)

• if \(y \le 1\) then suppose \(y \notin P\).

• then \(ax \le \beta\) for \(x \in P\) but \(ay > \beta\)

• since \(0 \in P\), \(\beta \ge 0\)

• if \(\beta > 0\) then \(\frac{a}{\beta}x \le 1\ \forall x \in P\) so its in \(P^*\) but \(\frac{a}{\beta}y > 1\) so it is a better opt for \(y\) contra

• if \(\beta = 0\) then \(\lambda ax \le 0 \le 1 \forall \lambda>0\) so \(\lambda a \in P^*\) but \(\lambda a y>1\) for some \(\lambda>0\) so is better opt for \(y\) contra.

Interior Point

Ellipsoid has problems in practice (\(O(n^6)\) for one). So people developed a different approach that has been extremely successful.

What goes wrong with simplex?

• follows edges of polytope

• complex stucture there, run into walls, etc

• interior point algorithms stay away from the walls, where structure simpler.

• Karmarkar did the first one (1984)

• huge media hype not accurate at the time

• now interior point is competitive with Simplex and often better

• arguably known from 1960s as “barrier methods”

• but visibility from Karmarkar set off huge progress

Primal-Dual Method

• Solve primal and dual together

• not by combining into one LP

• use current dual to tell you how to improve primal

• and vice versa

• (This idea has become very powerful in non-LP combinatorial optimization also)

Potential Reduction

Potential function:

• Idea: use a (nonlinear) potential function that is minimized at opt but also enforces feasibility

• use gradient descent to optimize the potential function.

• argue make “sufficient progress per step” to finish in poly steps

• Use logarithmic barrier function \[G(x) = cx-\mu(\sum\ln x_j)\] and try to minimize it

• first term aims for optimal objective

• second enforces positivity

• note barrier prevents from ever hitting optimum since \(G \rightarrow \infty\) at boundary

• but as discussed above ok to just get close.

• then “round” to a better vertex, which will be opt

Early methods used gradient descent:

• immediately take \(\mu\) small so objective dominates

• then use gradient descent to minimize \(G_\mu(x)\)

More recent, arguably cleaner approach is central path

• for each \(\mu\) there is an optimum point

• varying \(\mu\) traces out the central path

• Start with \(\mu\) huge, objective function term negligible

  • Equivalent to “start with a feasible point”

  • And requires just as much thinking as simplex, ellipsoid

  • but same kind of tricks work

• shrink \(\mu\) towards 0

• eventually objective dominates as \(\mu \rightarrow 0\)

• so we reach opt

• in the continuous domain unconcerned with runtime, this “obviously” works

• we need a discrete version

Characterizing the central path

• fix \(\mu\)

• central path minimizes \(G_\mu(x)\)

• imposes condition on gradient \(\nabla G(x) = \langle c_i-\mu/x_i \rangle\)

• but gradient may not be zero

• because we are constrained by \(Ax=b\)

• so real condition is that gradient is perpendicular to feasible set

• i.e. is a linear combination of rows of \(A\)

• (because these are the constraint hyperplanes’ normal vectors)

• i.e. \(yA = \langle c_i-\mu/x_i \rangle\)

• write \(yA+s=c\) where \(s_i =\mu/x_i > 0\) are slack variables

• so \(y,s\) is dual feasible

• duality gap is \(cx-yb = (yA+s)x - yb = sx = n\mu\)

• conversely, if have feasible \(x_i, s_i\) with \(x_is_i=\mu\), then we know we are on the central path at parameter \(\mu\)

• this is just saying each constraint is violated “equally” under proper scaling

How discretize?

• “predictor-corrector method”

• use linear approximation

• compute tangent to central path

• move along it as far as approximation holds (“predictor”)

• conclude \(\mu\) has improved on tangent and thus on corresponding central path point

• “correct” back to central path

• each step improves \(\mu\)

• stop when \(\mu\) “close enough” to 0

• I will show predictor step (moving towards better \(\mu\))

  • corrector step is similar but easier

  • just moving back to an optimum of the potential for new \(\mu\)

Termination

• as with ellipsoid, analyze time needed to get from start to “sufficiently good” \(\mu\)

• once duality gap smaller than bit-precision of vertices, done

  • as we saw before, can “round” to a vertex

  • since are closer to opt than second-best vertex, our starting objective is better than second best

  • so we can only round to opt

  • thus, sufficient to get \(\mu = 2^{-O(L)}\)

• as with simplex, there’s the cold-start problem

  • using similar method (designing related LP with easy starting point and opt at real desired starting point) can achieve \(\mu=2^L\)

• so solving only requires \(O(L)\) iterations of improving \(\mu\) by a constant

What is the improvement direction?

• From current \(x,s, \mu\), update to \(x+dx\), \(s+ds\) that are “on central path” for new, smaller \(\mu\)

• central path needs \((s_i+ds_i)(x_i+dx_i) = (\mu+d\mu)\) (same constant \(\forall i\))

  • i.e. \(s_i d(x_i) + d(s_i)x_i = d\mu +\) low order terms

• but also need \(A(dx)=0\) (to stay primal feasible)

• and \((dy)A+ds=0\) (to stay dual feasible)

• solve this linear system to get directions \(ds\) and \(dx\)

• these are the tangent for central path

Can we solve this?

• yes: if rescale, can see this as a projection problem

• rescale \(A\) so all \(x_i=1\), meaning all \(s_i = \mu\)

• equations now require \[\begin{aligned} \mu dx + ds & = &\mathbf{1}d\mu\\ Adx &= &0\\ (dy)A+ds &= &0\end{aligned}\]

• here \(\mathbf{1}\) is the all-ones vector

• in other words, \(\mu dx\) in nullspace of \(A\) and \(ds\) in span of \(A\)

• but they sum to \(\mathbf{1}d\mu\)

• so \(\mu dx\) and \(ds\) are just decomposition of \(\mathbf{1}d\mu\) onto \(A\) and \(A^{\perp}\)

• note this defines directions because both \(dx\) and \(ds\) (and \(d\mu\) and \(dy\)) can be multiplied by same arbitrary scalar

• move in this direction until approximation breaks

How far?

• Simplify

  • recall rescaled coordinates until all \(x_i=1\)

  • so all \(s_i = \mu\)

  • and duality gap is \(\sum s_i = n\mu\)

  • rescale \(dx_i\) and \(ds_i\) so “predicted” change in \(\mu\) is \(d\mu = \mu\)

  • i.e. linear approximation predicts each \(x_is_i\) decreases by \(\mu\)

  • predicts we close duality gap by moving \(dx,ds\)

• how far in that direction can we move—parameter \(\theta\)—before approximation breaks?

  • we want to be on central path, i.e. \((x_i+\theta dx_i)(s_i+\theta ds_i) = \mu+d\mu\)

  • actual change is \(\theta (x_ids_i + s_idx_i) +\theta^2 ds_idx_i\)

  • our linear system says these quantities equal for all \(i\) to first order (ignoring \(dx_ids_i\) term)

  • so until our linear approximation breaks down, all \(s_ix_i\) are changing by about same amount

  • i.e., we are near central path

• approximation breaks when second order term is comparable to first order

  • so want \(\theta^2dx_ids_i \ll \theta (x_ids_i + s_idx_i)\).

  • i.e. \(\theta \ll x_i/dx_i + s_i/ds_i\)

  • we rescaled \(x_i=1\) and \(s_i=\mu\), so need \(\theta \ll \mu/ds_i + 1/dx_i\).

• so how big can \(dx_i\) be?

  • remember we solved \(\mu dx + ds =\) orthogonal decomposition of \(d\mu\)

  • and rescaled so length of \(d\mu\) is \(\mu\)

  • i.e. \(\mu dx\) is projection of \(\mathbf{1}\mu\), so \(dx\) is projection of \(\mathbf{1}\)

  • all one’s vector has length \(\sqrt{n}\)

  • so any \(dx_i \le \sqrt{n}\)

  • similarly, any \(ds_i \le \mu\sqrt{n}\)

  • so requirement that \(\theta \ll \mu/ds_i+1/dx_i\) met by \(\theta \ll \mu/(\mu \sqrt{n}) + 1/\sqrt{n} = O(1/\sqrt{n})\)

  • so we can set \(\theta \approx 1/\sqrt{n}\) and have second-order term negligible

• How much improvement?

  • we said if \(\theta=1\) we close duality gap to zero (in linear approximation)

  • since we only move \(1/\sqrt{n}\) we shave a \(1/\sqrt{n}\) factor.

  • but that means we reduce \(\mu\) by a constant in \(\sqrt{n}\) steps

  • which means we finish in poly\((n)\) steps!

Interior point has recently been revolutionizing maxflow.