Linear programming duality

Every linear maximization program has a dual minimization program whose feasible objective values bound the primal values.

Minimizing ${\max }_i(a_i^{T}x+b_i)$ over unrestricted $x$ is a linear program after introducing an epigraph variable $t$ and inequalities $a_i^{T}x+b_i\le t$. Its optimization Lagrangian has finite infimum over $x,t$ exactly when the nonnegative multipliers sum to one and their weighted slopes sum to zero. The displayed Lagrangian dual problem maximizes the resulting constant lower bound. It identifies the role of a zero-slope convex combination of the affine functions.

For a minimization problem with inequalities $g_i(x)\le 0$, the Lagrangian is $L(x,\lambda )=f(x)+{\sum }_i{\lambda }_i{g}_i(x)$ with ${\lambda }_i\ge 0$. The Lagrangian dual maximizes the concave dual function $q(\lambda )={\inf }_{x}L(x,\lambda )$.

The Lagrange dual function is the infimum of a Lagrangian over its primal variables. It is concave in the multipliers, even before convexity of the primal problem is assumed. Maximizing it over sign-compatible multipliers gives a lower bound on a minimization problem by weak duality; appropriate convex qualifications give strong duality.

A Lagrangian relaxation moves selected constraints into the objective with appropriately signed Lagrange multipliers. Maximizing the relaxed objective over a larger easy set gives an upper bound for a constrained maximization problem. Optimizing that bound need not solve the original integer problem.

For binary knapsack optimization, $U(\lambda )$ is an upper bound for every $\lambda \ge 0$. Its minimum is computable from the finitely many profit-to-weight breakpoints and agrees with the fractional knapsack problem optimum. Zero-weight positive-profit terms are added independently.

For a primal maximization problem $Ax\le b$, $x\ge 0$, the dual minimizes $b^{T}y$ subject to $A^{T}y\ge c$, $y\ge 0$.

The dual pairsatisfies weak duality because $b^{T}y\le y^{T}Ax=x^T{A}^{T}y\le c^{T}x$ for every feasible pair.

Every feasible dual objective value bounds every feasible primal objective value.

For the linear program maximizing $c^{T}x$ with $Ax\le b$, $x\ge 0$, a vector $y\ge 0$ with $A^{T}y\ge c$ proves $c^{T}x\le y^{T}b$ for every feasible $x$. If a feasible $x$ attains that bound, it is optimal. This is weak duality written as a directly checkable certificate; adding nonnegative multiples of constraints suffices to verify it. The equivalent reversed-inequality certificate applies to a minimization program.

Strong duality means that the primal and dual optimal values are equal. It holds for every feasible bounded linear program and for broad classes of convex optimization problems under a constraint qualification.

A jointly proper convex function $f(x,z)$ represents an optimization problem at perturbation $z=0$ and a family of nearby problems at other $z$. Its primal value function is $p(z)={\inf }_{x}f(x,z)$. Relaxing an inequality by $z$ gives the example $f(x,z)=J(x)+{\delta }_{(-\infty ,0]}(r(x)-\sigma -z)$, using an indicator functional. The resulting convex perturbation duality expresses multipliers as supporting subgradients of the value function.

For a convex perturbation function, defineThe primal and dual values are $p(0)$ and $q(0)$; the signed dual marginal $q$ is concave. The identity $p^{\ast }(y)=f^{\ast }(0,y)$ gives $q(0)=p^{\ast \ast }(0)$. If $p$ is proper and finite near zero, subgradients at zero prove strong duality with dual attainment. In finite dimensions the relative-interior condition $0\in \mathrm{ri}(\mathrm{dom}p)$ also suffices, provided $p(0)$ is finite. This does not itself prove primal attainment.

An optimal dual variable $y^{\ast }\in \partial p(0)$ bounds the effect of perturbing a convex value function:If $p$ is finite convex near zero, $p^{\prime }(0;d)={\max }_{y\in \partial p(0)}⟨y,d⟩$. A singleton subdifferential gives differentiability and a first-order expansion. For an upper-bound constraint relaxed by $z$, the derivative equals the negative of the nonnegative Lagrange multiplier; relaxing the bound can decrease the minimum value.

The Slater condition requires a feasible point at which every nonlinear convex inequality is strict and every affine equality holds. For a convex problem it implies strong duality and attainment of the dual optimum under standard finiteness assumptions.

An exact maximum-violation penalty replaces inequalities $g_i(x)\le 0$ by the unconstrained objective $f(x)+M\max (0,g_1(x),\dots ,g_m(x))$. If an optimal dual multiplier is ${\lambda }^{\ast }$, every $M>\Vert {\lambda }^{\ast }{\Vert }_1$ makes every penalized minimizer feasible and optimal.

At a primal-dual optimum, each positive variable corresponds to a tight dual constraint and conversely for positive dual variables.

The transportation problem minimizes linear shipping cost while matching prescribed row supplies and column demands.

The assignment problem chooses a one-to-one pairing of jobs and agents minimizing their total cost. Its minimum-cost flow formulation puts unit supplies on job vertices and unit demands on agent vertices of a bipartite graph. Integral feasible flows are perfect matchings.

The Hungarian algorithm maintains feasible assignment dual potentials and a matching of equality edges. If the alternating search cannot augment, adjust reachable job and agent potentials by the smallest reduced cost leaving the search set. This preserves feasibility and existing matching equalities while exposing a new equality edge. A resulting perfect matching attains the dual lower bound.

These feasible potentials give the lower bound ${\sum }_i{\lambda }_i-{\sum }_j{\mu }_j$ on every assignment cost. An assignment using only equality edges attains this bound and is optimal by weak duality and complementary slackness.

The transportation polytope is the set of nonnegative matrices with fixed row and column sums.

The positive cells of a nondegenerate transportation basis form a spanning tree in the bipartite graph of supply and demand vertices.

The transportation simplex computes row and column potentials, enters a negative-reduced-cost cell, and pivots around its induced alternating cycle.

The northwest-corner method repeatedly allocates the smaller remaining supply and demand in the current top-left cell to construct an initial transportation basis.

For transportation potentials, the reduced cost is ${\bar{c}}_{ij}=c_{ij}-u_i-v_j$; a negative value identifies an improving entering cell.

Adding a nonbasic edge to a transportation tree creates one cycle; alternating equal additions and subtractions around it preserves every row and column sum.

Integer supplies and demands admit an integer optimum because an integer initial basis and every cycle pivot remain integer.

A matrix is totally unimodular when every square subdeterminant is $0$, $1$, or $-1$; linear programs with such a constraint matrix and integer right side have integral vertices.