The geometry of SDP-exactness in quadratic optimization

Cifuentes, Diego; Harris, Corey; Sturmfels, Bernd

doi:10.1007/s10107-019-01399-8

Cited by 19 publications

(55 citation statements)

References 16 publications

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“…Along with an invariancy set, a nonlinearity interval can be construed as a stability region and its identification has a high impact on the post-optimal analysis of SDO problems, see e.g., [9,10], in which the stability region of rank-one primal optimal solutions is of interest. Interestingly, the optimal value function for SDO problems has been shown to be piecewise algebraic [25], i.e., for each piece there exists a polynomial function Ψ(., .)…”

Section: Contributionsmentioning

confidence: 99%

Parametric analysis of semidefinite optimization

Mohammad-Nezhad

Terlaky

2019

Optimization

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In this paper, we study parametric analysis of semidefinite optimization problems w.r.t. the perturbation of the objective function. We study the behavior of the optimal partition and optimal set mapping on a so-called nonlinearity interval. Furthermore, we investigate the sensitivity of the approximation of the optimal partition in a nonlinearity interval, which has been recently studied by Mohammad-Nezhad and Terlaky. The approximation of the optimal partition was obtained from a bounded sequence of interior solutions on, or in a neighborhood of the central path. We derive an upper bound on the distance between the approximations of the optimal partitions of the original and perturbed problems. Finally, we examine the theoretical bounds by way of experimentation. KEYWORDSParametric semidefinite optimization; Optimal partition; Nonlinearity interval; Maximally complementary solution AMS CLASSIFICATION 90C51, 90C22, 90C25Assumption 2. The interior point condition holds at = 0, i.e., there exists a strictly feasible solution XLet E ⊆ R be the set of all for which v( ) > −∞. By Assumption 2, E is nonempty and nonsingleton, since both (P ) and (D ) have strictly feasible solutions for all in a sufficiently small neighborhood of 0. Further, v( ) is a proper concave function on E, and E is a closed, possibly unbounded, interval, see e.g., Lemma 2.2 in [4]. The continuity of v( ) on int(E) follows from its concavity on E, see Corollary 2.109 in [6].The primal and dual optimal set mappings are defined asOur analysis relies on the existence of central path and maximally complementary solutions [11], as formally defined below.Definition 1.1. We call an optimal solution X * ( ), y * ( ), S * ( ) maximally complementary if X * ( ) ∈ ri P * ( ) , and y * ( ), S * ( ) ∈ ri D * ( ) , where ri(.) denotes the relative interior of a convex set. An optimal solution X * ( ), y * ( ), S * ( ) is called strictly complementary if X * ( ) + S * ( ) 0.As a result of a theorem of the alternative [7], Assumption 2 implies the interior point condition at every ∈ int(E), see Lemma 3.1 in [14]. Therefore, strong duality 1 holds, and both P * ( ) and D * ( ) are nonempty and compact for all ∈ int(E), see e.g., Theorem 5.81 in [6]. Consequently, for every ∈ int(E) there exists a maximally complementary solution, and an optimal solution X( ), y( ), S( ) satisfies the complementarity condition X( )S( ) = 0. It is known that for an SDO problem, and in general a linear conic optimization problem, there might be no strictly complementary solution. Related worksSteady advances in computational optimization have enabled us to solve a wide variety of SDO problems in polynomial time. Nevertheless, sensitivity analysis tools are still the missing parts of SDO solvers, e.g., interior point methods (IPMs) in SeDuMi [32], SDPT3 [33,34], and MOSEK 2 . Shapiro [29] established the differentiability of the optimal solution for a nonlinear SDO problem using the standard implicit function theorem. Under linear perturbations in the objective vector, the coeffici...

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Section: Contributionsmentioning

confidence: 99%

Parametric analysis of semidefinite optimization

Mohammad-Nezhad

Terlaky

2019

Optimization

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“…To see that this subset is not closed, fix k = 2, n = 3 , let t be a parameter, and consider the quadrics q 1 = x 2 1 and q 2 = x 2 2 + tx 1 x 3 . Their span is a 2-dimensional subspace L t in 3 for all t ∈ ℝ .…”

Section: Proofmentioning

confidence: 99%

“…The first space L ∩ I 2 L is the linear span of the spectrahedon L ∩ n + , while the second space L ∩ I L also records tangent directions relative to the PSD cone. To illustrate the inclusions, we consider the bad plane in Example 1, where q = x 2 1 , I L = ⟨x 1 ⟩ and v = x 1 x 2 ∈ I L �I 2 L . We already know that the existence of such a pair (q, v) characterizes non-closed projections.…”

Section: Definitionmentioning

confidence: 99%

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Bad projections of the PSD cone

Jiang

Sturmfels

2021

Collect. Math.

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The image of the cone of positive semidefinite matrices under a linear map is a convex cone. Pataki characterized the set of linear maps for which that image is not closed. The Zariski closure of this set is a hypersurface in the Grassmannian. Its components are the coisotropic hypersurfaces of symmetric determinantal varieties. We develop the convex algebraic geometry of such bad projections, with focus on explicit computations.

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“…Cifuentes et al [2017] have recently shown a stability result implying that the semidefinite relaxation of Euclidean projection onto a smooth, quadratically-defined variety is exact in a neighborhood of the variety. Cifuentes et al [2018] have additionally shown that the region in which the relaxation is exact is a semialgebraic set, and they have provided a formula for the degree of its algebraic boundary. Unfortunately, computing this boundary is generally intractible for interesting varieties.…”

Section: Semidefinite Relaxationsmentioning

confidence: 99%

Algebraic Representations for Volumetric Frame Fields

2020

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Fig. 1. Octahedral fields generated with mMBO + RTR on various models.Field-guided parametrization methods have proven effective for quad meshing of surfaces; these methods compute smooth cross fields to guide the meshing process and then integrate the fields to construct a discrete mesh.A key challenge in extending these methods to three dimensions, however, is representation of field values. Whereas cross fields can be represented by tangent vector fields that form a linear space, the 3D analog-an octahedral frame field-takes values in a nonlinear manifold. In this work, we describe the space of octahedral frames in the language of differential and algebraic geometry. With this understanding, we develop geometry-aware tools for optimization of octahedral fields, namely geodesic stepping and exact projection via semidefinite relaxation. Our algebraic approach not only provides an elegant and mathematically-sound description of the space of octahedral frames but also suggests a generalization to frames whose three axes scale independently, better capturing the singular behavior we expect to see in volumetric frame fields. These new odeco frames, so-called as they are represented by orthogonally decomposable tensors, also admit a semidefinite program-based projection operator. Our description of the spaces of octahedral and odeco frames suggests computing frame fields via manifold-based optimization algorithms; we show that these algorithms efficiently produce high-quality fields while maintaining stability and smoothness.

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The geometry of SDP-exactness in quadratic optimization

Cited by 19 publications

References 16 publications

Parametric analysis of semidefinite optimization

Parametric analysis of semidefinite optimization

Bad projections of the PSD cone

Algebraic Representations for Volumetric Frame Fields

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