Exact duals and short certificates of infeasibility and weak infeasibility in conic linear programming

  title={Exact duals and short certificates of infeasibility and weak infeasibility in conic linear programming},
  author={Minghui Liu and G{\'a}bor Pataki},
  journal={Mathematical Programming},
In conic linear programming—in contrast to linear programming—the Lagrange dual is not an exact dual: it may not attain its optimal value, or there may be a positive duality gap. The corresponding Farkas’ lemma is also not exact (it does not always prove infeasibility). We describe exact duals, and certificates of infeasibility and weak infeasibility for conic LPs which are nearly as simple as the Lagrange dual, but do not rely on any constraint qualification. Some of our exact duals generalize… 

Strong duality and boundedness in conic optimization

Unlike linear programming, it is well-known that some conditions are required to achieve strong duality between a primal-dual pair of conic programs. The most common and well-known of these

An echelon form of weakly infeasible semidefinite programs and bad projections of the psd cone

A simple echelon form of weakly infeasible semidefinite SDPs is described with the following properties: it is obtained by elementary row operations and congruence transformations, it makes weak infeasibility evident, and it permits us to construct any weakly Infeasible SDP or bad linear projection by an elementary combinatorial algorithm.

Weak infeasibility in second order cone programming

This work considers a sequence of feasibility problems which mostly preserve the feasibility status of the original problem and shows that for a given weakly infeasible problem at most m directions are needed to get arbitrarily close to the cone.

Bad semidefinite programs with short proofs, and the closedness of the linear image of the semidefinite cone

Semidefinite programs (SDPs) -- some of the most useful and pervasive optimization problems of the last few decades -- often behave pathologically: the optimal values of the primal and dual problems

On strong duality, theorems of the alternative, and projections in conic optimization

A conic program is the problem of optimizing a linear function over a closed convex cone intersected with an affine preimage of another cone. We analyse three constraint qualifications, namely a

Characterizing Bad Semidefinite Programs: Normal Forms and Short Proofs

This work characterize pathological semidefinite systems by certain excluded matrices, which makes their pathological behavior easy to verify, and introduces readers to a fundamental issue in convex analysis: the linear image of a closed convex set may not be closed, and often simple conditions are available to verify the closedness, or lack of it.

On positive duality gaps in semidefinite programming

We present a novel analysis of semidefinite programs (SDPs) with positive duality gaps, i.e. different optimal values in the primal and dual problems. These SDPs are extremely pathological, often

Solving SDP completely with an interior point oracle

An analysis of double facial reduction, which is the process of applying facial reduction twice: first to the original problem and then once more to the dual of the regularized problem obtained during the first run.

Solving Conic Optimization Problems via Self-Dual Embedding and Facial Reduction: A Unified Approach

An algorithm based on facial reduction for solving the primal problem that, in principle, always succeeds and can be implemented by assuming oracle access to the central-path limit poi...

Necessary and Sufficient Conditions for Rank-One-Generated Cones

A closed convex conic subset [Formula: see text] of the positive semidefinite (PSD) cone is rank-one generated (ROG) if all of its extreme rays are generated by rank-one matrices. The ROG property of



Exact Duality in Semidefinite Programming Based on Elementary Reformulations

This work obtains an exact, short certificate of infeasibility in SDP by an elementary approach: it reformulate any equality constrained semidefinite system using only elementary row operations, and rotations, and concludes that when a system is infeasible, the reformulated system is trivially infeasable.

An exact duality theory for semidefinite programming and its complexity implications

In this paper, an exact dual is derived for Semidefinite Programming (SDP), for which strong duality properties hold without any regularity assumptions, and the dual is then applied to derive certain complexity results for SDP.

An Exact Duality Theory for Semidefinite Programming Based on Sums of Squares

This work provides nonlinear algebraic certificates for all infeasible linear matrix inequalities in the spirit of real algebraic geometry, and presents a new exact duality theory for semidefinite programming, motivated by the real radical and sums of squares certificates from real mathematics.

Strong Duality for Semidefinite Programming

The relationships among various duals are discussed and a unified treatment for strong duality in semidefinite programming is given.

Bad Semidefinite Programs: They All Look the Same

The main motivation is the striking similarity of badly behaved semidefinite systems in the literature; it characterize such systems by certain excluded matrices, which are easy to spot in all published examples.

Preprocessing and Regularization for Degenerate Semidefinite Programs

This paper presents a backward stable preprocessing technique for (nearly) ill-posed semidefinite programming, SDP, problems, i.e., programs for which the Slater constraint qualification (SCQ), the

Set Intersection Theorems and Existence of Optimal Solutions

It is shown how these conditions can be used to obtain simple and unified proofs of some known results on existence of optimal solutions, and to derive some new results, including a new extension of the Frank–Wolfe Theorem for (nonconvex) quadratic programming.

Strong Duality in Conic Linear Programming: Facial Reduction and Extended Duals

The facial reduction algorithm (FRA) of Borwein and Wolkowicz and the extended dual of Ramana provide a strong dual for the conic linear program $$\displaystyle{ \sup \,\{\,\langle c,x\rangle

How to generate weakly infeasible semidefinite programs via Lasserre’s relaxations for polynomial optimization

This note shows how to use Lasserre’s semidefinite programming relaxations for polynomial optimization in order to generate examples of weakly infeasible SDP and observes that SDP solvers do not detect the infeasibility and that values returned by SDPsolvers are equal to the optimal value of the instance due to numerical round-off errors.

Exact duality for optimization over symmetric cones

A strong duality theory for optimization problems over symmetric cones without assuming any constraint qualification is presented and it is argued that new software for homogeneous cone optimization problems should be developed.