mikedeng1

Solved 0

No accepted proofs yet.

Posted 50

• Theorem 10.7 — Strict Complementary Slackness TheoremOpen

  Oct 2026

• Theorem 10.6 — strictly complementary feasible solutionsOpen

  Oct 2026

• Theorem 10.4 — Separation Theorem for polyhedraOpen

  Oct 2026

• Lemma 10.5 — Farkas' LemmaProved

  Oct 2026

• Theorem 5.3 — Complementary Slackness TheoremProved

  Oct 2026

• Theorem 5.2 — Strong Duality TheoremProved

  Oct 2026

• Theorem 5.1 — Weak Duality TheoremProved

  Oct 2026

• Halfspaces (nonzero normal) and polyhedra in $\mathbb{R}^n$Definition

  Oct 2026

• The primal–dual pair $\max c^Tx,\ Ax+w=b$ / $\min b^Ty,\ A^Ty-z=c$: slacks, feasibility, optimalityDefinition

  Oct 2026

• Theorem 3.4 — fundamental theorem of linear programmingOpen

  Oct 2026

• Theorem 3.3 — the simplex method terminates under Bland's ruleOpen

  Oct 2026

• Theorem 3.2 — the simplex method terminates under the lexicographic ruleOpen

  Oct 2026

• Theorem 3.1 — a nonterminating simplex run must cycleOpen

  Oct 2026

• Feasible, optimal, infeasible, unbounded and basic solutions of a standard-form LPDefinition

  Oct 2026

• Simplex pivots, Bland's rule and the lexicographic ruleDefinition

  Oct 2026

• Dictionaries of a standard-form LP, determined by their basisDefinition

  Oct 2026

• Theorem 4.1 — the value function of decomposition with respect to variables is convex, with subgradient $g^x_{L_U}(\bar x, y(\bar x))$Open

  Oct 2026

• Theorem 4.3 — the Lagrangian dual bound $Q = \max_{u \ge 0} \Phi(u)$ does not exceed $f^*$Proved

  Oct 2026

• Theorem 4.2 — exactness of nonsmooth penalty functions with slopes above the Lagrange multipliersOpen

  Oct 2026

• Lemma 4.3 — the stochastic transportation problem is a convex programOpen

  Oct 2026

• Corollary of Theorem 4.1 — formula (4.7) for functions differentiable in $y$Open

  Oct 2026

• Theorem 4.1, formula (4.6) — the $x$-projection of a subgradient of $L_U$ with null $y$-projection is a subgradient of $\Phi$Proved

  Oct 2026

• Theorem 4.1, proof (p. 95) — $L_U$ has a subgradient at $(\bar x, y(\bar x))$ with null $y$-projectionOpen

  Oct 2026

• Theorem 4.1, proof (p. 95) — Kuhn–Tucker multipliers of the subproblem exist under Slater's conditionOpen

  Oct 2026

• Theorem 4.1 (first part) — the value function $\Phi$ is convex where it is definedProved

  Oct 2026

• The stochastic transportation problem (4.163)–(4.165)Definition

  Oct 2026

• Convex program (4.178): solution set, Lagrange multiplier vectors, nonsmooth penalty (4.179) and dual function (4.187)Definition

  Oct 2026

• Decomposition with respect to variables: the value function $\Phi$, Slater's condition, Kuhn–Tucker multipliers and partial subgradientsDefinition

  Oct 2026

• Theorem 3.14 — the space-dilation ellipsoid method keeps $x^*$ in $\{x : \|A_k(x_k - x)\| \le (n+1)h_k\}$Open

  Oct 2026

• p. 90 — the pseudo-gradient field of a convex–concave saddle point problemProved

  Oct 2026

• Eq. (3.65) — a vector field for the general convex programming problem (3.64)Proved

  Oct 2026

• Eq. (3.62) — a vector field for minimizing a convex function on a ballProved

  Oct 2026

• p. 87 — the localizing ellipsoids shrink in volume by the ratio $q_n < 1$ per iterationOpen

  Oct 2026

• p. 87 — the localizing ellipsoid $\Phi_k$ has volume $v_0 R^n (n/\sqrt{n^2-1})^{nk}/\det A_k$Proved

  Oct 2026

• Eq. (3.4) — the norm of a vector after space dilation along $\xi$Proved

  Oct 2026

• The space-dilation operator $R_\alpha(\xi)$ and Shor's ellipsoid algorithm (3.57)–(3.60)Definition

  Oct 2026

• Theorem 3.13 — the $r(\alpha)$-algorithm converges to an isolated local minimumOpen

  Oct 2026

• Theorem 3.12 — the level set $\{f = f_\infty\}$ contains a point with linearly dependent $G_f$Open

  Oct 2026

• Theorem 3.11 — along the $r(\alpha)$-algorithm, $p(\bar P_{\delta,\varepsilon}(x_k))$ is infinitely often at least $\sqrt{(v^2\alpha^{2/n}-1)/(\alpha^2-1)}$Open

  Oct 2026

• Lemma 3.3 — contracting along a long chord of $W$ keeps the width above $d(W)/\sqrt{1 + (1-\beta^2)/(\beta^2\gamma^2)}$Open

  Oct 2026

• Lemma 3.2 — the width of $BW$ lies between $\lambda(B)\,d(W)$ and $\lambda(B)\,D(W)$Open

  Oct 2026

• The class $K$ of piecewise smooth functions, $G_f(x)$, $\bar P_{\delta,\varepsilon}(x)$ and runs of the $r_\mu(\alpha)$-algorithmDefinition

  Oct 2026

• Space dilation $R_\alpha(\xi)$, widths $d_\eta(W)$, $d(W)$, $D(W)$ and the ratio $p(W)$ of a convex bodyDefinition

  Oct 2026

• Theorem 3.4 — SDG function values converge at the geometric rate $\alpha^{-k/n}$Open

  Oct 2026

• Theorem 3.3 — the SDG iterates satisfy $\|A_k(x_k - x^*)\| \le d$Open

  Oct 2026

• Theorem 3.2 — the record of $\|\tilde g_r\|$ is at most $d\sqrt{k(\alpha^2-1)}/\sqrt{\alpha^{2k/n}-1}$Open

  Oct 2026

• Theorem 3.1 — along a subsequence $\|\tilde g_{k_p}\| < c\,(\prod_{j\le k_p}\alpha_j)^{-1/n}$Open

  Oct 2026

• Eq. (3.4) — the norm of a dilated vectorProved

  Oct 2026

• The space-dilation operator $R_\alpha(\xi)$ and the subgradient method with space dilation along the gradient (SDG)Definition

  Oct 2026

• Theorem 2.5 — if $M^*$ contains a ball of radius $r$ and $\limsup h_k < 2r$, the method (2.4) terminatesProved

  Oct 2026