A Safety, Tolerability, and Pharmacokinetics Study of MLN0128 as a Single Agent and in Combination With Paclitaxel in Adults With Advanced Nonhematologic Malignancies

Inclusion Criteria: 1. Age is ≥ 18 years, including males and females. 2. Has Advanced nonhematologic malignancies, with the exception of primary brain tumor, and have failed or are not eligible for standard of care therapy. History of brain metastasis may be allowed if all the following criteria are met: brain metastases have been treated, there is no evidence of progression or hemorrhage after treatment, dexamethasone discontinued for ≥ 4 weeks before first study drug administration, and there is no ongoing requirement for dexamethasone or anti-epileptic drugs. 3. Has received not more than 4 prior lines of systemic cytotoxic chemotherapy for advanced or metastatic disease. 4. Has Eastern Cooperative Oncology Group (ECOG) Performance Status 0 to 1. 5. Has adequate organ function, including the following: * Bone marrow reserve consistent with absolute neutrophil count (ANC) ≥ 1.5 x 10\^9/L; platelet count ≥ 100 x 10\^9/L; and hemoglobin ≥ 9 g/dL without transfusion in the last 2 weeks * Hepatic: total bilirubin ≤ 1.5 x the upper limit of the normal range (ULN), transaminases (aspartate aminotransferase \[AST\]/serum glutamic oxaloacetic transaminase \[SGOT\] and alanine aminotransferase \[ALT\]/serum glutamic pyruvic transaminase \[SGPT\]) ≤ 2.5 x ULN (≤ 5 x ULN if liver metastases are present) * Renal: normal serum creatinine or calculated creatinine clearance ≥ 60 mL/min based either on Cockcroft-Gault estimate or based on urine collection (12- or 24-hour) * Metabolic: fasting serum glucose (≤ 130 mg/dL) and fasting triglycerides ≤ 300 mg/dL 6. Has left ventricular ejection fraction (LVEF) within 5 absolute percentage points of institutional standard of normal as measured by echocardiogram (ECHO) or multiple gated acquisition scan (MUGA) within 4 weeks before first study drug administration (ie, if the institutional normal is 50%, participant's LVEF may be as low as 45% to be eligible for the study). 7. Is a female participants who: * Are postmenopausal for at least 1 year before the screening visit, OR * Are surgically sterile, OR * If they are of childbearing potential, agree to practice 2 effective methods of contraception, at the same time, from the time of signing the informed consent form through 90 days after the last dose of study drug, or * Agree to practice true abstinence, when this is in line with the preferred and usual lifestyle of the participant. (Periodic abstinence (eg, calendar, ovulation, symptothermal, postovulation methods\] and withdrawal are not acceptable methods of contraception.) Male participants, even if surgically sterilized (ie, status postvasectomy), who: * Agree to practice effective barrier contraception during the entire study treatment period and through 90 days after the last dose of study drug, or * Agree to practice true abstinence, when this is in line with the preferred and usual lifestyle of the participant. (Periodic abstinence \[eg, calendar, ovulation, symptothermal, postovulation methods for the female partner\] and withdrawal are not acceptable methods of contraception.) 8. Has ability to swallow oral medications. 9. Has voluntary written consent obtained before performance of any study-related procedure not part of standard medical care, with the understanding that consent may be withdrawn by the participant at any time without prejudice to future medical care. Exclusion Criteria: 1. Has a diagnosis of primary brain tumor. 2. Has untreated brain metastasis or history of leptomeningeal disease or spinal cord compression. 3. Has failed to recover from the reversible effects of prior anticancer therapies with the exception of alopecia, and after-effects associated with prior tyrosine kinase inhibitor therapy, such as hair depigmentation, hypothyroidism, and/or splinter hemorrhage. 4. Has received prior cancer therapy or other investigational therapy within 2 weeks before the first administration of study drug. For prior therapies with a half-life longer than 3 days, the interval must be at least 28 days before the first administration of study drug, and the participant must have documented disease progression. 5. Has initiation of hematopoietic growth factors within 1 week before the first administration of any study drug; participants already receiving hematopoietic growth factors on a chronic basis for ≥ 4 weeks are eligible. 6. Has chronic systemic corticosteroid (except inhalers) use within 1 week before the first administration of study drug. Premedication with dexamethasone before paclitaxel administration in this study is allowed. 7. Has manifestations of malabsorption due to prior gastrointestinal surgery, gastrointestinal disease, or for an unknown reason that may alter the absorption of MLN0128. 8. Has poorly controlled diabetes mellitus defined as glycosylated hemoglobin (HbA1c) \> 7%; participants with a history of transient glucose intolerance due to corticosteroid administration are allowed if all other eligibility criteria are met. 9. Has other clinically significant comorbidities, such as uncontrolled pulmonary disease, active central nervous system disease, active infection, or any other condition that could compromise the participant's participation in the study. 10. Has a known human immunodeficiency virus infection. 11. Is pregnant (positive serum or urine pregnancy test) or breastfeeding. 12. Has any history of unstable angina, myocardial infarction, New York Heart Association Class III or IV heart failure, and/or pulmonary hypertension. 13. Has significant active cardiovascular disease including: * Uncontrolled high blood pressure (ie, systolic blood pressure \> 180 mmHg, diastolic blood pressure \> 95 mmHg) * Grade 3 or higher valvular disease * Grade 3 or higher atrial fibrillation * Grade 3 or higher bradycardia * Endocarditis * Pulmonary embolism * Recent cerebrovascular accident/transient ischemic attack ≤ 6 months before enrollment 14. Diagnosed or treated for another malignancy within 2 years before the first dose or previously diagnosed with another malignancy and have any evidence of residual disease. Participants with nonmelanoma skin cancer or carcinoma in situ of any type are not excluded if they have undergone complete resection. 15. Single-Agent QD Arm participants participating in the pharmacokinetic (PK) Run-In (only): participants who use proton pump inhibitors (PPIs) less than 5 days before the first MLN0128 PK Run-In dose OR use H2 receptor antagonists within 24 hours of the first PK Run-In dose.