Convex Duality Research Articles

Explicit A Posteriori Error Representation for Variational Problems and Application to TV-Minimization

AbstractIn this paper, we propose a general approach for explicit a posteriori error representation for convex minimization problems using basic convex duality relations. Exploiting discrete orthogonality relations in the space of element-wise constant vector fields as well as a discrete integration-by-parts formula between the Crouzeix–Raviart and the Raviart–Thomas element, all convex duality relations are transferred to a discrete level, making the explicit a posteriori error representation –initially based on continuous arguments only– practicable from a numerical point of view. In addition, we provide a generalized Marini formula that determines a discrete primal solution in terms of a given discrete dual solution. We benchmark all these concepts via the Rudin–Osher–Fatemi model. This leads to an adaptive algorithm that yields a (quasi-optimal) linear convergence rate.

The support of mixed area measures involving a new class of convex bodies

Mixed volumes in n-dimensional Euclidean space are functionals of n-tuples of convex bodies K,L,C1,…,Cn−2. The Alexandrov–Fenchel inequalities are fundamental inequalities between mixed volumes of convex bodies. As very special cases they cover or imply many important inequalities between basic geometric functionals. A complete characterization of the equality cases in the Alexandrov–Fenchel inequality remains a challenging open problem. Major recent progress was made by Yair Shenfeld and Ramon van Handel [13,14], in particular they resolved the problem in the cases where C1,…,Cn−2 are polytopes, zonoids or smooth bodies (under some dimensional restriction). In [6] we introduced the class of polyoids, which are defined as limits of finite Minkowski sums of polytopes having a bounded number vertices. Polyoids encompass polytopes, zonoids and triangle bodies, and they can be characterized by means of generating measures. Based on this characterization and Shenfeld and van Handel's contribution (and under a dimensional restriction), we extended their result to polyoids (or smooth bodies). Our previous result was stated in terms of the support of the mixed area measure associated with the unit ball Bn and C1,…,Cn−2. This characterization result is completed in the present work which more generally provides a geometric description of the support of the mixed area measure of an arbitrary (n−1)-tuple of polyoids (or smooth bodies). The result thus (partially) confirms a long-standing conjecture by Rolf Schneider in the case of polyoids, and hence in particular it covers the case of zonoids and triangle bodies.

Quantified Legendreness and the Regularity of Minima

We introduce a new quantification of nonuniform ellipticity in variational problems via convex duality, and prove higher differentiability and 2d-smoothness results for vector valued minimizers of possibly degenerate functionals. Our framework covers convex, anisotropic polynomials as prototypical model examples—in particular, we improve in an essentially optimal fashion Marcellini’s original results (Marcellini in Arch Rat Mech Anal 105:267–284, 1989).

Optimizing local electricity markets: A bi-level primal-dual approach for integrating price-based demand response and distribution locational marginal pricing

This paper addresses the conundrum of optimizing electricity consumption patterns in response to fluctuations in demand and price, a task managed by both load aggregators and the distribution system operator (DSO). The conventional approaches in the literature to integrate demand response (DR) into optimal power flow (OPF) problems typically overlook the price responsiveness of consumers or simplify power flow equations to account for price-elastic demand.In this paper, we strive to close this gap by introducing a bi-level primal-dual optimization framework that incorporates aggregators' objectives into the market-clearing process while preserving the precision of the OPF equations. At the upper level, the model seeks to minimize both the total payment and peak load. The lower level, structured as a second-order conic (SOC) problem, aims to reduce overall generation costs subject to constraints of the second-order conic (SOC) branch flow model (BFM). The two levels interact through the optimal demand response and distribution locational marginal prices (DLMPs). The principles of convex duality principles together with the strong duality constraint are leveraged to transform the market-clearing problem into a single primal-dual problem. We also circumvent the non-linearity that stems from the DR payment term by incorporating discretizing loads and the big-M method, thus converting the problem into a mixed-integer SOC (MI-SOC) formulation. The merits of the proposed MI-SOC framework are validated through case studies conducted on the IEEE 33-bus test system, showcasing its potential to enforce price-elasticity demand response in distribution system's electricity markets.

On some algorithms for estimation in Gaussian graphical models

Abstract In Gaussian graphical models, the likelihood equations must typically be solved iteratively. This paper investigates two algorithms: a version of iterative proportional scaling, which avoids inversion of large matrices, and an algorithm based on convex duality and operating on the covariance matrix by neighbourhood coordinate descent, which corresponds to the graphical lasso with zero penalty. For large, sparse graphs, the iterative proportional scaling algorithm appears feasible and has simple convergence properties. The algorithm based on neighbourhood coordinate descent is extremely fast and less dependent on sparsity, but needs a positive-definite starting value to converge. We provide an algorithm for finding such a starting value for graphs with low colouring number. As a consequence, we also obtain a simplified proof of existence of the maximum likelihood estimator in such cases.

Proof-Theoretical Aspects of Nonlinear and Set-Valued Analysis

AbstractThis thesis is concerned with extending the underlying logical approach as well as the breadth of applications of the proof mining program to various (mostly previously untreated) areas of nonlinear analysis and optimization, with a particular focus being placed on topics which involve set-valued operators.For this, we extend the current logical methodology of proof mining by new systems and corresponding so-called logical metatheorems that cover these more involved areas of nonlinear analysis. Most of these systems crucially rely on the use of intensional methods, treating sets with potentially high quantifier complexity in the defining matrix via characteristic functions and axioms that describe only their properties and do not completely characterize the elements of the sets.The applicability of all of these metatheorems is then substantiated by a range of case studies for the respective areas which in particular also highlight the naturalness of the use of intensional methods in the design of the corresponding systems.The first new area covered thereby is the theory of nonlinear semigroups induced by corresponding evolution equations for accretive operators. In that context, we present (besides an initial foray into the area from 2015) essentially the first applications of proof mining to the theory of partial differential equations. Concretely, we provide quantitative versions of four central results on the asymptotic behavior of solutions to such equations.The second new area unlocked in this thesis is that of the continuous dual of a Banach space and its norm (which are also approached via intensional methods). This in particular relies on a proof-theoretically tame treatment of suprema over (certain) bounded sets in this intensional context which is further exploited later on. These systems, which give access to this until now untreated fundamental notion from functional analysis, are then used to provide further substantial extensions to treat various notions from convex analysis like the Fréchet derivative of a convex function, Fenchel conjugates, Bregman distances, and monotone operators on Banach spaces in the sense of Browder.These systems are then utilized to provide applications in the context of Picard- and Halpern-style iterations of so-called Bregman strongly nonexpansive mappings where we provide both new quantitative and qualitative results.Lastly, we discuss the key notion of extensionality of a set-valued operator and its relation to set-theoretic maximality principles in more depth (which was already singled out—to some degree—in previous work). We thereby exhibit an issue arising with treating full extensionality in the context of these intensional approaches to set-valued operators and present useful fragments of the full extensionality statement where these issues are avoided.Corresponding to these fragments, we discuss a range of uniform continuity statements for set-valued operators beyond the usual notion involving the Hausdorff-metric. In particular, in that context, we utilize the previous tame treatment of suprema over bounded sets to also provide the first proof-theoretic treatment of that Hausdorff-metric in the context of systems for proof mining.The applicability of this treatment of the Hausdorff-metric is then in particular substantiated by a last case study where we provide quantitative information for a Mann-type iteration of set-valued mappings which are nonexpansive w.r.t. the Hausdorff-metric.The abstract was taken directly from the thesis.Abstract prepared by Nicholas Pischke, Technische Universität Darmstadt, Darmstadt, GermanyE-mail: pischke@mathematik.tu-darmstadt.de.URL: https://doi.org/10.26083/tuprints-00026584.

A Notion of Fenchel Conjugate for Set-Valued Mappings

A Notion of Fenchel Conjugate for Set-Valued Mappings

ALM for insurers with multiple underwriting lines and portfolio constraints: a Lagrangian duality approach

We investigate how an insurance firm can optimally allocate its assets to back up liabilities from multiple non-life business lines. The insurance risks are modeled by a multidimensional jump-diffusion process that accounts for simultaneous claims in different insurance lines with policy limits. We use Lagrangian convex duality techniques to derive optimal investment-underwriting strategies that maximize the expected utility from dividends and final wealth over a finite horizon. We examine how risk aversion, prudence, portfolio constraints, and multivariate insurance risk affect the firm’s earnings retention. We obtain explicit solutions for optimal strategies under constant relative risk aversion preferences. Finally, we illustrate our results with numerical examples and show the impact of co-integration for asset-liability management with multiple sources of insurance risk.

Stability from rigidity via umbilicity

Abstract We consider a range of geometric stability problems for hypersurfaces of spaceforms. One of the key results is an estimate relating the distance to a geodesic sphere of an embedded hypersurface with integral norms of the traceless Hessian operator of a level set function for the open set bounded by the hypersurface. As application, we give a unified treatment of many old and new stability problems arising in geometry and analysis. Those problems ask for spherical closeness of a hypersurface, given a geometric constraint. Examples include stability in Alexandroff’s soap bubble theorem in space forms, Serrin’s overdetermined problem, a Steklov problem involving the bi-Laplace operator and non-convex Alexandroff–Fenchel inequalities.

Michael-Simon type inequalities in hyperbolic space H n + 1 ${\mathbb{H}}^{n+1}$ via Brendle-Guan-Li’s flows

Abstract In the present paper, we first establish and verify a new sharp hyperbolic version of the Michael-Simon inequality for mean curvatures in hyperbolic space H n + 1 ${\mathbb{H}}^{n+1}$ based on the locally constrained inverse curvature flow introduced by Brendle, Guan and Li (“An inverse curvature type hypersurface flow in H n + 1 ${\mathbb{H}}^{n+1}$ ,” (Preprint)) as follows (0.1) ∫ M λ ′ f 2 E 1 2 + | ∇ M f | 2 − ∫ M ∇ ̄ f λ ′ , ν + ∫ ∂ M f ≥ ω n 1 n ∫ M f n n − 1 n − 1 n $$\underset{M}{\int }{\lambda }^{\prime }\sqrt{{f}^{2}{E}_{1}^{2}+\vert {\nabla }^{M}f{\vert }^{2}}-\underset{M}{\int }\langle \bar{\nabla }\left(f{\lambda }^{\prime }\right),\nu \rangle +\underset{\partial M}{\int }f\ge {\omega }_{n}^{\frac{1}{n}}{\left(\underset{M}{\int }{f}^{\frac{n}{n-1}}\right)}^{\frac{n-1}{n}}$$ provided that M is h-convex and f is a positive smooth function, where λ′(r) = coshr. In particular, when f is of constant, (0.1) coincides with the Minkowski type inequality stated by Brendle, Hung, and Wang in (“A Minkowski inequality for hypersurfaces in the anti-de Sitter-Schwarzschild manifold,” Commun. Pure Appl. Math., vol. 69, no. 1, pp. 124–144, 2016). Further, we also establish and confirm a new sharp Michael-Simon inequality for the kth mean curvatures in H n + 1 ${\mathbb{H}}^{n+1}$ by virtue of the Brendle-Guan-Li’s flow (“An inverse curvature type hypersurface flow in H n + 1 ${\mathbb{H}}^{n+1}$ ,” (Preprint)) as below (0.2) ∫ M λ ′ f 2 E k 2 + | ∇ M f | 2 E k − 1 2 − ∫ M ∇ ̄ f λ ′ , ν ⋅ E k − 1 + ∫ ∂ M f ⋅ E k − 1 ≥ p k ◦ q 1 − 1 ( W 1 ( Ω ) ) 1 n − k + 1 ∫ M f n − k + 1 n − k ⋅ E k − 1 n − k n − k + 1 \begin{align}\hfill & \underset{M}{\int }{\lambda }^{\prime }\sqrt{{f}^{2}{E}_{k}^{2}+\vert {\nabla }^{M}f{\vert }^{2}{E}_{k-1}^{2}}-\underset{M}{\int }\langle \bar{\nabla }\left(f{\lambda }^{\prime }\right),\nu \rangle \cdot {E}_{k-1}+\underset{\partial M}{\int }f\cdot {E}_{k-1}\hfill \\ \hfill & \quad \ge {\left({p}_{k}{\circ}{q}_{1}^{-1}\left({W}_{1}\left({\Omega}\right)\right)\right)}^{\frac{1}{n-k+1}}{\left(\underset{M}{\int }{f}^{\frac{n-k+1}{n-k}}\cdot {E}_{k-1}\right)}^{\frac{n-k}{n-k+1}}\hfill \end{align} provided that M is h-convex and Ω is the domain enclosed by M, p k (r) = ω n (λ′) k−1, W 1 ( Ω ) = 1 n | M | ${W}_{1}\left({\Omega}\right)=\frac{1}{n}\vert M\vert $ , λ′(r) = coshr, q 1 ( r ) = W 1 S r n + 1 ${q}_{1}\left(r\right)={W}_{1}\left({S}_{r}^{n+1}\right)$ , the area for a geodesic sphere of radius r, and q 1 − 1 ${q}_{1}^{-1}$ is the inverse function of q 1. In particular, when f is of constant and k is odd, (0.2) is exactly the weighted Alexandrov–Fenchel inequalities proven by Hu, Li, and Wei in (“Locally constrained curvature flows and geometric inequalities in hyperbolic space,” Math. Ann., vol. 382, nos. 3–4, pp. 1425–1474, 2022).

A multi-marginal c-convex duality theorem for martingale optimal transport

A convex duality result for martingale optimal transport problems with two marginals was established in Beiglböck et al. (2013). In this paper we provide a generalization of this result to the multi-period setting.

A complete family of Alexandrov–Fenchel inequalities for convex capillary hypersurfaces in the half-space

A complete family of Alexandrov–Fenchel inequalities for convex capillary hypersurfaces in the half-space

Mean-variance portfolio selection with non-linear wealth dynamics and random coefficients

This paper studies the continuous time mean-variance portfolio selection problem with one kind of non-linear wealth dynamics. To deal with the expectation constraint, an auxiliary stochastic control problem is firstly solved by two new generalized stochastic Riccati equations from which a candidate portfolio in feedback form is constructed, and the corresponding wealth process will never cross the vertex of the parabola. In order to verify the optimality of the candidate portfolio, the convex duality (requires the monotonicity of the cost function) is established to give another more direct expression of the terminal wealth level. The variance-optimal martingale measure and the link between the non-linear financial market and the classical linear market are also provided. Finally, we obtain the efficient frontier in closed form. From our results, people are more likely to invest their money in riskless asset compared with the classical linear market.

Equality cases of the Alexandrov–Fenchel inequality are not in the polynomial hierarchy

Abstract Describing the equality conditions of the Alexandrov–Fenchel inequality [Ale37] has been a major open problem for decades. We prove that in the case of convex polytopes, this description is not in the polynomial hierarchy unless the polynomial hierarchy collapses to a finite level. This is the first hardness result for the problem and is a complexity counterpart of the recent result by Shenfeld and van Handel [SvH23], which gave a geometric characterization of the equality conditions. The proof involves Stanley’s [Sta81] order polytopes and employs poset theoretic technology.

A framework for safe decision making: A convex duality approach

We study the problem of online interaction in general decision making problems, where the objective is not only to find optimal strategies, but also to satisfy certain safety guarantees, expressed in terms of costs accrued. In particular, we focus on the online learning problem in which an agent has to find the optimal solution of a linear objective. Moreover, the agent has to satisfy a linear safety constraint at each round. We propose a theoretical framework to address such problems and present BAN-SOLO, a UCB-like algorithm that, in an online interaction with an unknown environment, attains sublinear regret of order O ( T ) and satisfies a safety constraint with high probability at each iteration. BAN-SOLO provides a general framework that can be applied to any setting in which estimators of the objective and the cost function are available. At its core, it relies on tools from convex duality to manage environment exploration while satisfying the safety constraint imposed by the problem. To show the applicability of our framework, we provide two game theoretical applications: normal-form games and sequential decision-making problems.

Grid entropy in last passage percolation — A superadditive critical exponent approach

Working in the setting of i.i.d. last-passage percolation on RD with no assumptions on the underlying edge-weight distribution, we arrive at the notion of grid entropy — a Subadditive Ergodic Theorem limit of the entropies of paths with empirical measures weakly converging to a given target, or equivalently a deterministic critical exponent of canonical order statistics associated with the Levy-Prokhorov metric. This provides a fresh approach to an entropy first developed by Rassoul-Agha and Seppäläinen as a large deviation rate function of empirical measures along paths. In their 2014 paper, variational formulas are developed for the point-to-point/point-to-level Gibbs Free Energies as the convex conjugates of this entropy. We rework these formulas in our new framework and explicitly link our descriptions of grid entropy to theirs. We also improve on a known bound for this entropy by introducing a relative entropy term in the inequality. Furthermore, we show that the set of measures with finite grid entropy coincides with the deterministic set of limit points of empirical measures studied in a 2020 paper by Bates. We partially answer a directed polymer version of a question of Hoffman which was previously tackled in the zero temperature case by Bates.

Correntropy-Based Low-Rank Matrix Factorization With Constraint Graph Learning for Image Clustering.

This article proposes a novel low-rank matrix factorization model for semisupervised image clustering. In order to alleviate the negative effect of outliers, the maximum correntropy criterion (MCC) is incorporated as a metric to build the model. To utilize the label information to improve the clustering results, a constraint graph learning framework is proposed to adaptively learn the local structure of the data by considering the label information. Furthermore, an iterative algorithm based on Fenchel conjugate (FC) and block coordinate update (BCU) is proposed to solve the model. The convergence properties of the proposed algorithm are analyzed, which shows that the algorithm exhibits both objective sequential convergence and iterate sequential convergence. Experiments are conducted on six real-world image datasets, and the proposed algorithm is compared with eight state-of-the-art methods. The results show that the proposed method can achieve better performance in most situations in terms of clustering accuracy and mutual information.

A dual approach to agency problems

This paper presents a dual approach to the standard model of moral hazard. We formulate the dual of the principal–agent problem under the assumption that the incentive constraint can be replaced by a local constraint (the first-order approach), to examine whether the relaxed agency problem yields a candidate solution. The dual formulation generates a convex conjugate, which transforms the agent’s utility from compensation into a dual functional. The dual problem features a simple convex structure, which enables us to perform a comprehensive analysis for the agency problem. We derive novel and more tractable conditions for existence and uniqueness of a solution to the problem with the dual elements. Furthermore, the approach to the dual problem provides illuminating insights into the previous nonexistence results.

Fenchel Conjugate via Busemann Function on Hadamard Manifolds

Fenchel Conjugate via Busemann Function on Hadamard Manifolds

Robust enhanced indexation optimization with sparse industry layout constraint

In this paper, we investigate an enhanced indexation methodology using robust Conditional Value-at-Risk (CVaR) and group-sparse optimization. A featured difference from the existing literatures is to describe the tail risk using the worst-case CVaR of excess returns, and the process of industry selection using a weighted ℓ∞,1-norm constraint. We develop an accelerated alternating minimization algorithm for solving this problem. At each iteration, this method usually alternately solves a convex cone program, which admits a closed-form solution via convex duality theory, and a projection problem onto a weighted infinity-to-one-ball, where a fixed-point iteration projection method is developed, terminating in finite number of iterations. The global convergence rates in terms of the primal and dual residuals are also provided. Empirical tests on actual data sets are presented to demonstrate the superior out-of-sample performance of our proposed strategy.