Polynomial Optimization Research Articles

Color Dominance-Based Polynomial Optimization Segmentation for Identifying Tomato Leaves and Fruits

Optimization processes or methods play an essential role in the continuous improvement of various human activities, particularly in agriculture, given its vital role in food production. In precision agriculture, which utilizes technology to optimize food production, a primary goal is to minimize the consumption of resources like water, fertilizers, and the detection of pests and diseases. In the fertilization process, it is essential to identify any deficiencies or excesses of chemical elements. Nutrient deficiencies, which are essential for plant development, are typically detected in the leaves of crops. This paper proposes a methodology for optimizing the color threshold dominance factors employed in the segmentation process for tomato crop leaves and fruits. The optimization is performed using an interpolation method to find the values that maximize the segmentation of leaves and fruits used by the color dominance segmentation method. A comparison of the interpolation method results with those obtained using a greedy algorithm, which iteratively finds the optimal segmentation values, shows nearly identical outcomes. Similarly, a UNetmodel is used for semantic segmentation, the results of which are inferior to those obtained by the proposed interpolation optimization method. The most significant contribution of the interpolation method is that it requires only a single iteration to generate the initial data, in contrast to the iterative search required by the greedy algorithm and the lengthy training process and video card dependency of the UNet model. This results in an 80% reduction in computation time.

A real moment-HSOS hierarchy for complex polynomial optimization with real coefficients

A real moment-HSOS hierarchy for complex polynomial optimization with real coefficients

A Lagrange barrier approach for the minimum concave cost supply problem via a logarithmic descent direction algorithm

The minimisation of concave costs in the supply chain presents a challenging non-deterministic polynomial (NP) optimisation problem, widely applicable in industrial and management engineering. To approximate solutions to this problem, we propose a logarithmic descent direction algorithm (LDDA) that utilises the Lagrange logarithmic barrier function. As the barrier variable decreases from a high positive value to zero, the algorithm is capable of tracking the minimal track of the logarithmic barrier function, thereby obtaining top-quality solutions. The Lagrange function is utilised to handle linear equality constraints, whilst the logarithmic barrier function compels the solution towards the global or near-global optimum. Within this concave cost supply model, a logarithmic descent direction is constructed, and an iterative optimisation process for the algorithm is proposed. A corresponding Lyapunov function naturally emerges from this descent direction, thus ensuring convergence of the proposed algorithm. Numerical results demonstrate the effectiveness of the algorithm.

Generalized Semi-infinite Polynomial Optimization and Semidefinite Programming Relaxations

Generalized Semi-infinite Polynomial Optimization and Semidefinite Programming Relaxations

Monomial asymptotic polynomials and applications to polynomial optimization problems

The problem of polynomial optimization plays an important role in many fields such as physics, chemistry, and economics. This problem has received research attention from many mathematicians recently. In this paper, we study the polynomial optimization problem over a non-compact semi-algebraic set, for which its constraint set of polynomials G is asymptotic with a finite family of monomials. By changing variables via a suitable monomial mapping, we transform the problem under consideration into the polynomial optimization problem over a compact semi-algebraic feasible set. We then apply the well-known result that the optimal value of a polynomial over a compact semi-algebraic set can be approximated as closely as desired by solving a hierarchy of semi-definite programs and the convergence is finite generically, to obtain results in the general case when the cone C(G) is unimodular. In particular, in the case of polynomials in two variables, we solve the problem quite completely without requiring C(G) to be unimodular

Solving clustered low-rank semidefinite programs arising from polynomial optimization

We study a primal-dual interior point method specialized to clustered low-rank semidefinite programs requiring high precision numerics, which arise from certain multivariate polynomial (matrix) programs through sums-of-squares characterizations and sampling. We consider the interplay of sampling and symmetry reduction as well as a greedy method to obtain numerically good bases and sample points. We apply this to the computation of three-point bounds for the kissing number problem, for which we show a significant speedup. This allows for the computation of improved kissing number bounds in dimensions 11 through 23. The approach performs well for problems with bad numerical conditioning, which we show through new computations for the binary sphere packing problem.

Stable recovery of piecewise constant conductance on spider networks

ABSTRACT We address the discrete inverse conductance problem for well-connected spider networks, that is, to recover the conductance function on a well-connected spider network from the Dirichlet-to-Neumann map. It is well-known that this inverse problem is exponentially ill-posed, requiring the implementation of a regularization strategy for numerical solutions. Our focus lies in exploring whether prior knowledge of the conductance being piecewise constant within a partition of the edge set comprising a few subsets enables stable conductance recovery. To achieve this, we propose formulating the problem as a polynomial optimization problem, incorporating a regularization term that accounts for the piecewise constant hypothesis. We show several experimental examples in which the stable conductance recovery under the aforementioned hypothesis is feasible.

Polynomial Optimization Over Unions of Sets

AbstractThis paper studies the polynomial optimization problem whose feasible set is a union of several basic closed semialgebraic sets. We propose a unified hierarchy of Moment-SOS relaxations to solve it globally. Under some assumptions, we prove the asymptotic or finite convergence of the unified hierarchy. Special properties for the univariate case are discussed. The application for computing (p, q)-norms of matrices is also presented.

A Moment-Sum-of-Squares Hierarchy for Robust Polynomial Matrix Inequality Optimization with Sum-of-Squares Convexity

We study a class of polynomial optimization problems with a robust polynomial matrix inequality (PMI) constraint where the uncertainty set itself is also defined by a PMI. These can be viewed as matrix generalizations of semi-infinite polynomial programs because they involve actually infinitely many PMI constraints in general. Under certain sum-of-squares (SOS)-convexity assumptions, we construct a hierarchy of increasingly tight moment-SOS relaxations for solving such problems. Most of the nice features of the moment-SOS hierarchy for the usual polynomial optimization are extended to this more complicated setting. In particular, asymptotic convergence of the hierarchy is guaranteed, and finite convergence can be certified if some flat extension condition holds true. To extract global minimizers, we provide a linear algebra procedure for recovering a finitely atomic matrix-valued measure from truncated matrix-valued moments. As an application, we are able to solve the problem of minimizing the smallest eigenvalue of a polynomial matrix subject to a PMI constraint. If SOS convexity is replaced by convexity, we can still approximate the optimal value as closely as desired by solving a sequence of semidefinite programs and certify global optimality in case that certain flat extension conditions hold true. Finally, an extension to the nonconvexity setting is provided under a rank 1 condition. To obtain the above-mentioned results, techniques from real algebraic geometry, matrix-valued measure theory, and convex optimization are employed. Funding: This work was supported by the National Key Research and Development Program of China [Grant 2023YFA1009401] and the National Natural Science Foundation of China [Grants 11571350 and 12201618].

The pseudo-Boolean polytope and polynomial-size extended formulations for binary polynomial optimization

The pseudo-Boolean polytope and polynomial-size extended formulations for binary polynomial optimization

Hierarchy relaxations for robust equilibrium constrained polynomial problems and applications to electric vehicle charging scheduling

In this paper, we consider a polynomial problem with equilibrium constraints in which the constraint functions and the equilibrium constraints involve data uncertainties. Employing a robust optimization approach, we examine the uncertain equilibrium constrained polynomial optimization problem by establishing lower bound approximations and asymptotic convergences of bounded degree diagonally dominant sum-of-squares (DSOS), scaled diagonally dominant sum-of-squares (SDSOS) and sum-of-squares (SOS) polynomial relaxations for the robust equilibrium constrained polynomial optimization problem. We also provide numerical examples to illustrate how the optimal value of a robust equilibrium constrained problem can be calculated by solving associated relaxation problems. Furthermore, an application to electric vehicle charging scheduling problems under uncertain discharging supplies shows that for the lower relaxation degrees, the DSOS, SDSOS and SOS relaxations obtain reasonable charging costs and for the higher relaxation degrees, the SDSOS relaxation scheme has the best performance, making it desirable for practical applications.

On symmetry adapted bases in trigonometric optimization

The problem of computing the global minimum of a trigonometric polynomial is computationally hard. We address this problem for the case, where the polynomial is invariant under the exponential action of a finite group. The strategy is to follow an established relaxation strategy in order to obtain a converging hierarchy of lower bounds. Those bounds are obtained by numerically solving semi-definite programs (SDPs) on the cone of positive semi-definite Hermitian Toeplitz matrices, which is outlined in the book of Dumitrescu Dumitrescu (2007). To exploit the invariance, we show that the group has an induced action on the Toeplitz matrices and prove that the feasible region of the SDP can be restricted to the invariant matrices, whilst retaining the same solution. Then we construct a symmetry adapted basis tailored to this group action, which allows us to block-diagonalize invariant matrices and thus reduce the computational complexity to solve the SDP.The approach is in its generality novel for trigonometric optimization and complements the one that was proposed as a poster at the ISSAC 2022 conference Hubert et al. (2022) and later extended to Hubert et al. (2024). In the previous work, we first used the invariance of the trigonometric polynomial to obtain a classical polynomial optimization problem on the orbit space and subsequently relaxed the problem to an SDP. Now, we first make the relaxation and then exploit invariance.Partial results of this article have been presented as a poster at the ISSAC 2023 conference Metzlaff (2023).

Evolving scientific discovery by unifying data and background knowledge with AI Hilbert

The discovery of scientific formulae that parsimoniously explain natural phenomena and align with existing background theory is a key goal in science. Historically, scientists have derived natural laws by manipulating equations based on existing knowledge, forming new equations, and verifying them experimentally. However, this does not include experimental data within the discovery process, which may be inefficient. We propose a solution to this problem when all axioms and scientific laws are expressible as polynomials and argue our approach is widely applicable. We model notions of minimal complexity using binary variables and logical constraints, solve polynomial optimization problems via mixed-integer linear or semidefinite optimization, and prove the validity of our scientific discoveries in a principled manner using Positivstellensatz certificates. We demonstrate that some famous scientific laws, including Kepler’s Law of Planetary Motion and the Radiated Gravitational Wave Power equation, can be derived in a principled manner from axioms and experimental data.

Multirate time-integration based on dynamic ODE partitioning through adaptively refined meshes for compressible fluid dynamics

In this paper, we apply the Paired-Explicit Runge-Kutta (P-ERK) schemes by Vermeire et al. [1,2] to dynamically partitioned systems arising from adaptive mesh refinement. The P-ERK schemes enable multirate time-integration with no changes in the spatial discretization methodology, making them readily implementable in existing codes that employ a method-of-lines approach.We show that speedup compared to a range of state of the art Runge-Kutta methods can be realized, despite additional overhead due to the dynamic re-assignment of flagging variables and restricting nonlinear stability properties. The effectiveness of the approach is demonstrated for a range of simulation setups for viscous and inviscid convection-dominated compressible flows for which we provide a reproducibility repository.In addition, we perform a thorough investigation of the nonlinear stability properties of the Paired-Explicit Runge-Kutta schemes regarding limitations due to the violation of monotonicity properties of the underlying spatial discretization. Furthermore, we present a novel approach for estimating the relevant eigenvalues of large Jacobians required for the optimization of stability polynomials.

Existence results of solutions to a generalized vertical polynomial complementarity problem in terms of vertical block tensor tuples

The goal of this paper is to investigate a class of vertical polynomial complementarity problems (denoted by VPCP), which is a subclass of complementarity problems. Firstly, we give a necessary and sufficient condition for the existence of solution for the VPCP. Secondly, the existence of the least element solution to the VPCP is shown when the involved tensor tuple is a vertical block Z-tensor tuple and the feasible set of the problem is nonempty. Particularly, we can find the least element solution of this problem by solving a polynomial optimization problem. In addition, some examples are given to demonstrate the validity of our findings.

T-semidefinite programming relaxation with third-order tensors for constrained polynomial optimization

T-semidefinite programming relaxation with third-order tensors for constrained polynomial optimization

Uncertainty Relations from State Polynomial Optimization.

Uncertainty relations are a fundamental feature of quantum mechanics. How can these relations be found systematically? Here, we develop a semidefinite programming hierarchy for additive uncertainty relations in the variances of noncommuting observables. Our hierarchy is built on the state polynomial optimization framework, also known as scalar extension. The hierarchy is complete in the sense that it converges to tight uncertainty relations. We improve upon upper bounds for all 1292 additive uncertainty relations on up to nine operators for which a tight bound is not known. The bounds are dimension-free and depend entirely on the algebraic relations among the operators. The techniques apply to a range of scenarios, including Pauli, Heisenberg-Weyl, and fermionic operators, and generalize to higher order moments and multiplicative uncertainty relations.

Certifying long-range quantum correlations through routed Bell tests

Losses in the transmission channel, which increase with distance, pose a major obstacle to photonics demonstrations of quantum nonlocality and its applications. Recently, Chaturvedi, Viola, and Pawlowski (CVP) [arXiv:2211.14231] introduced a variation of standard Bell experiments with the goal of extending the range over which quantum nonlocality can be demonstrated. These experiments, which we call `routed Bell experiments', involve two distant parties, Alice and Bob, and allow Bob to route his quantum particle along two possible paths and measure it at two distinct locations – one near and another far from the source. The premise is that a high-quality Bell violation in the short-path should constrain the possible strategies underlying the experiment, thereby weakening the conditions required to detect nonlocal correlations in the long-path. Building on this idea, CVP showed that there are certain quantum correlations in routed Bell experiments such that the outcomes of the remote measurement device cannot be classically predetermined, even when its detection efficiency is arbitrarily low. In this paper, we show that the correlations considered by CVP, though they cannot be classically predetermined, do not require the transmission of quantum systems to the remote measurement device. This leads us to define and formalize the concept of `short-range' and `long-range' quantum correlations in routed Bell experiments. We show that these correlations can be characterized through standard semidefinite-programming hierarchies for non-commutative polynomial optimization. We then explore the conditions under which short-range quantum correlations can be ruled out and long-range quantum nonlocality can be certified in routed Bell experiments. We point out that there exist fundamental lower-bounds on the critical detection efficiency of the distant measurement device, implying that routed Bell experiments cannot demonstrate long-range quantum nonlocality at arbitrarily large distances. However, we do find that routed Bell experiments allow for reducing the detection efficiency threshold necessary to certify long-range quantum correlations. The improvements, though, are significantly smaller than those suggested by CVP's analysis.

Mutually unbiased bases: polynomial optimization and symmetry

A set of k orthonormal bases of Cd is called mutually unbiased if |⟨e,f⟩|2=1/d whenever e and f are basis vectors in distinct bases. A natural question is for which pairs (d,k) there exist k mutually unbiased bases in dimension d. The (well-known) upper bound k≤d+1 is attained when d is a power of a prime. For all other dimensions it is an open problem whether the bound can be attained. Navascués, Pironio, and Acín showed how to reformulate the existence question in terms of the existence of a certain C∗-algebra. This naturally leads to a noncommutative polynomial optimization problem and an associated hierarchy of semidefinite programs. The problem has a symmetry coming from the wreath product of Sd and Sk.We exploit this symmetry (analytically) to reduce the size of the semidefinite programs making them (numerically) tractable. A key step is a novel explicit decomposition of the Sd≀Sk-module C([d]×[k])t into irreducible modules. We present numerical results for small d,k and low levels of the hierarchy. In particular, we obtain sum-of-squares proofs for the (well-known) fact that there do not exist d+2 mutually unbiased bases in dimensions d=2,3,4,5,6,7,8. Moreover, our numerical results indicate that a sum-of-squares refutation, in the above-mentioned framework, of the existence of more than 3 MUBs in dimension 6 requires polynomials of total degree at least 12.

Probabilistic failure envelopes of tripod bucket-supported offshore wind turbines in spatially variable clay based on a novel continuous method

Marine geologic conditions play key roles in offshore geotechnical preliminary design and construction decisions. Tripod bucket foundations are considered promising geotechnical foundations for offshore engineering. Their bearing characteristics are often determined based on tripod foundation system failure modes in uniform clay without considering spatially variable soil properties. Therefore, this study investigates the combined bearing capacity of tripod bucket foundations in spatially variable clay based on comprehensive finite element analysis with Monte Carlo simulation (MCS) using a novel simplified modified swipe (SMS) method considering the effects of bucket spacing and foundation skirt length. The SMS method can directly represent the ultimate combined bearing capacity of the whole space efficiently and accurately. The results show that a reduced bucket spacing or increased skirt length changes the failure mode of the foundation from independent caisson failure to global failure. Moreover, the spatial variation in soil leads to an asymmetric failure mode, which is more remarkable for foundations with small bucket group effects. An independent safety factor design approach is adopted to represent failure envelopes with different recurrence probabilities, which should be cost-effective, especially for foundations sensitive to soil variability. Finally, an application based on a convex polynomial optimization strategy for representing failure envelopes is demonstrated.