arXiv subjects

Boshi Yang

Publications and source records attributed to Boshi Yang.

4 recordsLinked to original sources

arXiv2026

Quadratic Convexification of a Square Truncated by a Hyperbola

We study the quadratic convexification of the compact nonconvex set ${G} := \left[1/2,2\right]^2 \cap \{(x_1,x_2)\in\mathbb R^2\mid x_1x_2\leq1\}, $ which is a square truncated by a hyperbolic arc. Although the ordinary convex hull of ${G}$ is a triangle, its lifted convex hull in the complete quadratic space retains the nontrivial geometry of the curved boundary. We characterize all extreme rays of the cone of quadratic polynomials nonnegative on $G$, including parameterized families of bounded tangent and bitangent rays. Using this classification and conic duality, we derive an exact finite semidefinite representation of the lifted convex hull of ${G}$. The analysis follows the general framework of our earlier work on an unbounded product-constrained region, but the bounded geometry creates new boundary-contact patterns and leads to a different finite organization of the nonnegative-quadratic cone.

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arXiv2026

Nonnegative Quadratics over a Quadrant with a Bilinear Constraint

We study quadratic polynomials that are nonnegative on the non-compact set \[ F:=\{(x_1,x_2)\in\mathbb R^2:\ x_1\ge 0,\ x_2\ge 0,\ x_1x_2\le 1\}. \] All extreme rays of the cone of nonnegative quadratic polynomials are characterized on this set. The characterization allows us to study parameterized valid inequalities for quadratic convexifications involving $F$, which yields a semidefinite representation of its lifted convex hull in the quadratic space. Our analysis on extreme ray characterization separates the positive-semidefinite (PSD) and non-PSD branches, reduces the latter to boundary nonnegativity, and classifies the boundary contacts of the relevant extreme rays. By reparameterization, we also find a non-trivial and non-permutation-symmetric six-dimensional linear section of the cone of nonnegative homogeneous ternary octics that are sum-of-squares. We also show that our lifted convex hull result yields a degree-bounded preordering certificate of nonnegative quadratics on $F$ and a degree-bounded certificate for a family of nonnegative quartics on the half-strip.

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arXiv2026

Pricing Discrete and Nonlinear Markets With Semidefinite Relaxations

Nonconvexities in markets with discrete decisions and nonlinear constraints make efficient pricing challenging, often necessitating subsidies. A prime example is the unit commitment (UC) problem in electricity markets, where costly subsidies are commonly required. We propose a new pricing scheme for nonconvex markets with both discreteness and nonlinearity, by convexifying nonconvex structures through a semidefinite programming (SDP) relaxation and deriving prices from the relaxation's dual variables. When the choice set is bounded, we establish strong duality for the SDP, which allows us to extend the envelope theorem to the value function of the relaxation. This extension yields a marginal price signal for demand, which we use as our pricing mechanism. We demonstrate that under certain conditions-for instance, when the relaxation's right hand sides are linear in demand-the resulting lost opportunity cost is bounded by the relaxation's optimality gap. This result highlights the importance of achieving tight relaxations. The proposed framework applies to nonconvex electricity market problems, including for both direct current and alternating current UC. Our numerical experiments indicate that the SDP relaxations are often tight, reinforcing the effectiveness of the proposed pricing scheme. Across a suite of IEEE benchmark instances, the lost opportunity cost under our pricing scheme is, on average, 46% lower than that of the commonly used fixed-binary pricing scheme.

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arXiv2024

Affine Facial Reduction for Semidefinite Relaxations of Binary and Mixed-Binary Optimization Problems

Semidefinite programming (SDP) relaxations can provide strong bounds for binary and mixed-binary optimization problems, but their practical use is often limited by matrix variables of large order and by failures of Slater's condition, which can contribute to numerical difficulties for interior-point solvers. We propose \emph{affine facial reduction (affine FR)}, an automatic, LP-based preprocessing method for SDP relaxations of such problems. The method uses the affine hull of the linear programming relaxation to construct a positive semidefinite face containing the lifted feasible set and an associated facial range vector, thereby replacing the original positive semidefinite matrix variable by one of lower order. The required affine-hull information is obtained by solving a single linear programming problem, followed by linear-algebraic computations. We analyze the relation between affine FR, analytical facial reduction, the Permenter--Parrilo partial FR method, and Sieve-SDP, and establish explicit comparisons among the resulting matrix orders for the Shor relaxation. On MIPLIB 2017 mixed-binary instances, affine FR reduces matrix order more often and by a larger amount than the partial-FR variants considered, with comparable preprocessing times. On selected SAT relaxations with substantial reductions, affine FR can shorten SDP solution times and yield more reliable solver outcomes when the unreduced formulations encounter numerical difficulties.

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